Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 482305 10 pageshttpdxdoiorg1011552013482305

Research ArticleOn the Formal Integrability Problem forPlanar Differential Systems

Antonio Algaba1 Cristoacutebal Garciacutea1 and Jaume Gineacute2

1 Departamento de Matematicas Facultad de Ciencias Avda Tres de Marzo sn 21071 Huelva Spain2Departament de Matematica Universitat de Lleida Avda Jaume II 69 25001 Lleida Catalonia Spain

Correspondence should be addressed to Jaume Gine ginematematicaudlcat

Received 15 November 2012 Accepted 28 January 2013

Academic Editor Sung Guen Kim

Copyright copy 2013 Antonio Algaba et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study the analytic integrability problem through the formal integrability problem and we show its connection in some caseswith the existence of invariant analytic (sometimes algebraic) curves From the results obtained we consider some families ofanalytic differential systems in C2 and imposing the formal integrability we find resonant centers obviating the computation ofsome necessary conditions

1 Introduction

One of the main open problems in the qualitative theoryof differential systems in R2 is the distinction between acenter and a focus called the center problem and its relationwith the integrability problem see for instance [1ndash5] Thenotion of center can be extended to the case of a 119901

minus119902 resonant singular point of a polynomial vector field inC2 and to some other situations (resonant node saddlenode and nonelementary singular points) see [6] Recentlyseveral works about this subject have appeared where theclassification of the resonant centers for certain familiesis given using powerful computational facilities see forinstance [6ndash17]

There exist several methods to find necessary conditionssee [1 18] However there is no general method to providethe sufficiency for each family that satisfies some necessaryconditions The sufficiency is obtained verifying that thesystem is Hamiltonian or that it has certain reversibility orcertain Lie symmetry or finding a first integral well definedin a neighborhood of the singular point sometimes findingan integrating factor that allows to construct a first integralreducing the system to an integrable system see for instance[4 5 18ndash23] and the references therein These methods have

proved ineffective in certain families that already verify somenecessary conditions and in many papers some cases areestablished as open problems see for instance [7 10 11]Carrying in practice to prove the sufficiency by ad hocmethods for each particular case where it has been possiblesee [6ndash8 12 16 17 24ndash26] One of the ways to prove thesufficiency is proving the existence of a formal first integralThe existence of a formal first integral implies the existenceof an analytic first integral for an isolated singularity fromthe results obtained by Mattei and Moussu see [27] andTheorem 2 in Section 2

Therefore one of the main objectives for the next yearswill be to find an algorithm (if exists) or methods that givedirectly sufficient conditions to decide whether or not adifferential system in the plane admits a formal first integralat a singular point

This paper is the first step in this direction In thispaper we study the analytic integrability through the formalintegrability and we show its connection in some caseswith the existence of invariant analytic (sometimes algebraic)curves Moreover from the results obtained in this workwe consider some families of analytic differential systems inC2 and imposing the formal integrability we find centers or

2 Abstract and Applied Analysis

resonant centers obviating the computation of the necessaryconditions (see for instance Proposition 11)

2 Definitions and Preliminary Concepts

We first consider a real system of ordinary differentialequations onR2 with an isolated singular point at the originwhose linear parts are nonzero pure imaginary numbers Bya linear change of coordinates and a constant time rescalingthe system takes the form

= V + sdot sdot sdot V = minus119906 + sdot sdot sdot (1)

The classical Poincare-Liapunov center theorem states thatthe origin is a center if and only if the system admits ananalytic first integral of the form 120601(119906 V) = 119906

2+ V2 + sdot sdot sdot see

for instance [18 28ndash31] and the references thereinIf system (1) is complexified in a natural way taking

119911 = 119906 + 119894V then we obtain a differential equation of theform = 119894119911 + sdot sdot sdot In this case one constructs step bystep the formal first integral Φ = 119911119911 + sdot sdot sdot satisfying theequation Φ = V

3|119911|4+ V5|119911|6+ sdot sdot sdot where the coefficients

V2119894+1

called the focus quantities are polynomials in thecoefficients of the original system The theorem of Poincare-Liapunov [32 33] says that the point 119911 = 0 is a center ifand only if all the V

2119894+1= 0 Existence of a first integral 120601

is equivalent to existence of an analytic first integral for thecomplexified equation of the form Φ = 119911119911 + sdot sdot sdot Takingthe complex conjugated equation there arises an analyticsystem of ordinary differential equations on C2 of the form = 119894119911 + sdot sdot sdot = minus119894119908 + sdot sdot sdot where 119908 = 119906 minus 119894V Hence after thecomplexification the system is transformed into an analyticsystem with eigenvalues +119894 and minus119894 This is the 1 minus1 resonantsingular point and the numbers V

2119894+1become the coefficients

of the resonant terms in its orbital normal form Dulac [34]chosen this way to approach the center problem for quadraticsystems

The next natural generalization is to consider the case ofan analytic vector field inC2 with 119901 minus119902 resonant elementarysingular point

= 119901119909 + sdot sdot sdot = minus119902119910 + sdot sdot sdot (2)

where 119901 119902 isin Z with 119901 119902 gt 0 These facts motivate thegeneralization of the concept of real center to certain classesof systems of ordinary differential equations on C2 In thiscase we have the following definition of a resonant center orfocus coming from Dulac [34] see also [6]

Definition 1 A 119901 minus119902 resonant elementary singular point ofan analytic system is a resonant center if and only if thereexists a local meromorphic first integral Φ = ℎ

0+ sdot sdot sdot with

ℎ0= 119909119902119910119901 This singular point is a resonant focus of order 119896

if and only if there is a formal power series Φ = 119909119902119910119901+ sdot sdot sdot

with the property Φ = 119892119896ℎ119896+1

0+ sdot sdot sdot

Recently several works are focused on the study ofresonant centers for complex analytic systems In theseworksthe integrability and linearizability problems are studied

The linearizability problem is focused on the study of theexistence of an analytic change of coordinates that linearizethe complex analytic system

Mattei and Moussu [27] proved the next result for allisolated singularities

Theorem 2 Assume that system (1) or (2) with an isolatedsingularity at the origin has a formal first integral 119867 isin

R[[119909 119910]] around it Then there exists an analytic first integralaround the singularity

In the light of the former result we can conclude that inorder to prove the existence of an analytic first integral weonly need to prove the existence of a formal first integralTherefore the formal integrability problem takes a primaryrole for the upcoming investigations on the center andresonant center problem

3 Blowup of Resonant Saddles

In this section we will consider the first method to approachthe formal integrability problem for resonant centers Weconsider the resonant analytic system (2) We now do theblowup of this singularity This means that we apply the map(119909 119910) rarr (119909 119911) = (119909 119910119909) The point 119909 = 119910 = 0 is replacedby the line 119909 = 0 which contains two singular points thatcorrespond to the separatrices of (2) and are the saddles 119901

1

which is (119901 + 119902) minus119901 resonant and 1199012which is (119901 + 119902) minus119902

resonant The following lemma is proved in [12] using thenormal orbital form of a resonant analytic system (2)

Lemma 3 If one of the points 1199011or 1199012is orbitally analytically

linearizable then the point 119909 = 119910 = 0 is a 119901 minus119902 resonantcenter

As a corollary of this lemma we get the following usefulproperty which can be used to approach the resonant centerproblem If the point 119901

1is orbitally analytically linearizable

then it has a formal first integral in the variables (119909 119911) of theform

119867 = 119909119901119911119902+119901

infin

sum

119899=0

119891119899(119909) 119911119899 (3)

with 119891119899(0) = 0 Hence the original 119901 minus119902 resonant point has

a nonformal first integral of the form

119909minus119902119910119902+119901

infin

sum

119899=0

119891119899(119909) 119909minus119899119910119899=

infin

sum

119899=0

119891119899(119909) 119909minus119902minus119899

119910119902+119901+119899

(4)

Consequently if the resonant point 119901 minus119902 has a non formalfirst integral of the form (4) then by Lemma 3 the resonantpoint 119901 minus119902 is a resonant center The same result can beestablished for the other saddle119901

2These types of results were

also given by Bruno in [35 36]

Abstract and Applied Analysis 3

4 The 120576-Method for Resonant Centers

We recall in this section the 120576-method developed in [23]which we apply here to resonant centers We consider system(2) which we write into the form

= 119875 (119909 119910) = 119901119909 + 1198651(119909 119910)

= 119876 (119909 119910) = minus119902 119910 + 1198652(119909 119910)

(5)

where 1198651(119909 119910) and 119865

2(119909 119910) are analytic functions without

constant and linear terms defined in a neighborhood of theorigin

To implement the algorithm we introduce a rescalingof the variables and a time rescaling given by (119909 119910 119905) rarr

(120576119901119909 120576119902119910 120576119903119905) where 120576 gt 0 and 119901 119902 and 119903 isin Z and system

(5) takes the form

= 120576119903minus119901

(119901 120576119901119909 + 1198651(120576119901119909 120576119902119910))

= 120576119903minus119902

(minus119902 120576119902119910 + 1198652(120576119901119909 120576119902119910))

(6)

We choose 119901 119902 119903 in such a way that system (6) will beanalytic in 120576 Hence by the classical theorem of the analyticdependence with respect to the parameters we have thatsystem (6) admits a first integral which can be developedin power of series of 120576 because it is analytic with respectto this parameter Therefore we can propose the followingdevelopment for the first integral

119867(119909 119910) =

infin

sum

119896=0

120576119896ℎ119896(119909 119910) (7)

where ℎ119896(119909 119910) are arbitrary functionsWenotice that119865

1(119909 119910)

and 1198652(119909 119910) are analytic functions where both can be null

and 1198651(0 0) = 119865

2(0 0) = 0 so we can develop them in a

neighborhood of the origin as convergent series of 119909 and 119910

of the form

1198651(119909 119910) = 119901

119899(119909 119910) + 119901

119899+1(119909 119910) + sdot sdot sdot + 119901

119895(119909 119910) + sdot sdot sdot

1198652(119909 119910) = 119902

119899(119909 119910) + 119902

119899+1(119909 119910) + sdot sdot sdot + 119902

119895(119909 119910) + sdot sdot sdot

(8)

with 119899 = minsubdeg|(00)

1198651(119909 119910) subdeg

|(00)1198652(119909 119910) ge 2

We recall that given an analytic function 119891(119909 119910)

defined in a neighborhood of a point (1199090 1199100) we define

subdeg|(1199090 1199100)

119891(119909 119910) as the least positive integer 119895 such thatsome derivative (120597119895119891120597119909119894120597119910119895minus119894)(119909

0 1199100) is not zero We notice

that this computation depends on the variables (119909 119910) onwhich the function 119891(119909 119910) depends so we will explicit thevariables used in each computation of subdeg For instancesubdeg

|(1199090 1199100)119891(119909 119910) = 0 if and only if 119891(119909

0 1199100) = 0 In (8)

119901119895(119909 119910) and 119902

119895(119909 119910) denote homogeneous polynomials of 119909

and 119910 of degree 119895 ge 119899 It is possible that 119901119899(119909 119910) or 119902

119899(119909 119910)

be null but by definition not both of them can be null Thesimplest case is to consider in the rescaling 119901 = 119902 = 1 Infact this case is equivalent to impose that system (8) has afirst integral which can be expanded as a formal series inhomogeneous parts

The richness of the 120576-method is that using the parameter120576 the functions ℎ

119896(119909 119910) need not be homogeneous parts

and we can construct also a singular series expansion in thevariables 119909 and 119910 see [23] The method depends heavily onℎ0the first integral of the initial quasi-homogeneous system

The simpler is ℎ0farther we go with the method However ℎ

0

can be chosen using different scalings of variables (119909 119910 119905) rarr

(120576119901119909 120576119902119910 120576119903119905) where 120576 gt 0 and 119901 119902 and 119903 isin Z in such

a way that ℎ0will be as simple as possible The method

gives necessary conditions to have analytic integrability ora singular series expansion around a singular point andinformation about what is called in [23] the essential variablesof a system In the method developed in [23] the parameter 120576needs not be small The parameter 120576 may be relatively large(for instance 120576 rarr 1) The convergence of series (7) withrespect to (119909 119910) must be analyzed in each particular caseand the convergent rate depends upon the nonlinear termsof the system (8) In the case of a resonant center the mostconvenient ℎ

0is 119909119902119910119901

From now on we suppose that system (5) admits aformal first integral Therefore ℎ

119896(119909 119910) are homogeneous

polynomials of degree 119896 and we can take ℎ0(119909 119910) = 0 In this

case if we impose that series (7) be a formal first integral ofsystem (5) we obtain at each order of homogeneity

119901119909120597ℎ119896

120597119909(119909 119910) minus 119902119910

120597ℎ119896

120597119910(119909 119910) + 119866

119896(119909 119910) = 0 (9)

where119866119896(119909 119910) depends on the previous ℎ

119894for 119894 = 1 119896minus1

The solution of the partial equation (9) is

ℎ119896(119909 119910) = 119865 (119909

119902119901119910) minus int

119909 119866119896(119904 (119909119902119901

119910) 119904119902119901

)

119901 119904119889119904 (10)

where 119865 is an arbitrary function of 119909119902119901119910 We require that ℎ119896

be a polynomial and therefore an ℎ119896not having logarithmic

terms In order to avoid the logarithmic terms it is easy to seethat the function119866

119896(119909 119910)must not have polynomial terms of

the form (119909119902119910119901)119898 for 119898 isin N This result is the same as that

we obtain using normal form theory see [6]We must impose the vanishing of all the logarithmic

terms (if there exists any difference from zero) and prove thatall the functions ℎ

119894(119909 119910) for 119894 gt 1 are polynomials In fact

we must prove this by induction assuming that all the ℎ119894for

119894 = 1 119896 minus 1 are polynomials and prove that the recursivepartial differential equation with respect to ℎ

119896(see (9)) gives

also a polynomial This is not so easy and even less workingwith a partial recursive differential equation as (9)

At this point we think in the blow-up 119911 = 119910119909 becauseeach ℎ

119894(119909 119910) are homogeneous polynomials of degree 119894

4 Abstract and Applied Analysis

This blowup transforms the system (5) into a system ofvariables (119909 119911) of the form

= minus (119901 + 119902) 119911 + 119909F (119909 119911)

= 119901119909 + 1199092G (119909 119911)

(11)

where F(0 0) = 0 After that we propose a formal firstintegral of the form

=

infin

sum

119894ge1

119891119894(119911) 119909119894 (12)

where 119891119894(119911) must be polynomials of degree lei (if the log-

arithmic terms are zero) If we impose that series (12) bea first integral of the system in variables (119909 119911) we obtainthat the term 119887

201199092 must be zero However we can vanish

this term doing the change 119910 = 119910 minus 1205721199092 and 119909 = 119909 with

120572 = minus11988720(2119901 + 119902) We must also take 119891

1(119911) = 0 Finally at

each power of 119910 we have the recursive equation

119896119901119891119896(119911) minus (119901 + 119902) 119911119891

1015840

119896(119911) + 119892

119896(119911) = 0 (13)

where 119892119896(119911) depends of the previous functions 119891

119894for 119894 =

1 119896 minus 1 Hence in this case instead of getting the partialrecursive differential equation (9) we obtain an ordinaryrecursive differential equation whose solution is given by

119891119896(119911) =

119896119911119896119901(119901+119902)

+ 119911119896119901(119901+119902)

int

119911 119904minus1minus119896119901(119901+119902)

119901 + 119902119892119896(119904) 119889119904

(14)

where 119896is an arbitrary constant However we are going to

see that although we place in a resonant center where all the119891119896be polynomials this blowup does not allow us to prove in

general by induction of the existence of a formal first integralof the form (12)

In this sense consider the recursive differential equation(13) whose solution is (14) and we assume that system (5) hasa resonant degenerate center at the origin It is easy to see thatthere always exists a value 119896

0such that for 119896 ge 119896

0the arbitrary

polynomials 119891119894(119911) for 119894 = 1 119896

0minus 1 can gives following the

recursive equation (13) logarithmic terms Therefore we cannot apply the induction method to prove that the recursiveequation (13) gives always a polynomial

To see this we must to see that the solution (14) of therecursive equation (13) can give a logarithmic term Thishappens when

minus1 minus1198960119901

119901 + 119902+ 119898119896= minus1 (15)

where119898119896is the degree of the polynomial 119892

119896(119904) From here we

have that 1198960= 119898119896(119901+119902)119901 Hence if119901 = 1 then 119896

0= 119898119896(1+119902)

which can be satisfied because119898119896and 119902 are positive integers

For the case 119901 = 1 and taking into account that the value of119898119896increases when 119896 increases it can also exist a value of119898

119896

such that 119898119896is divisible by 119901 and that gives the value of 119896

0

that can give logarithmic terms

Therefore the conclusion is that the formal constructionof the first integral (7) using homogeneous terms or using theblow-up 119911 = 119910119909 and the formal series (12) do not allow touse the induction method in order to verify the existence of aformal first integral We must use other developments whichis the subject of the next section

5 Other Developments of the FormalFirst Integral

In this section we consider other developments of the formalfirst integral We consider the formal development of the firstintegral of system (5) in a series in the variable 119909 or in 119910 thatis we consider

1198671=

infin

sum

119896=0

119891119896(119909) 119910119896 or 119867

2=

infin

sum

119896=0

119892119896(119910) 119909119896 (16)

First we consider a general analytic system that we canalways write into the following forms

= 119891 (119909) + 119910Φ1(119909 119910) = 119892 (119909) + 119910Φ

2(119909 119910) (17)

or

= 119891 (119910) + 119909Ψ1(119909 119910) = 119892 (119910) + 119909Ψ

2(119909 119910) (18)

where 119891 119892Φ1Φ2 Ψ1 and Ψ

2are analytic in their variables

For systems (17) and (18) we have the following straightfor-ward result

Proposition 4 If we impose that the series 1198671(1198672 resp) be

a first integral of system (17) (system (18) resp) among othersthe first condition is 1198911015840

0(119909)119891(119909) + 119891

1(119909)119892(119909) = 0 (1198911015840

0(119910)119892(119910) +

1198911(119910)119891(119910) = 0 resp)

In order that this condition generates a collection ofrecursive differential equations where each 119891

119894(119909) does not

depend on119891119894+1

(119909) wemust impose119892(119909) = 0 (119891(119910) = 0 resp)which implies that119910 = 0 (119909 = 0 resp) is an invariant algebraiccurve of system (17) (system (18) resp)

In fact for system (5) there are always a new coordinates(1199111 1199112) where 119911

1= 0 and 119911

2= 0 are invariant curves These

invariant curves are defined by the stable and instable man-ifold of the p minusq resonant singular point and therefore theprevious conditions are directly satisfied In these coordinatesany 119901 minus119902 resonant singular point is a Lotka-Volterra systemIn the following result we are going to see that we always canfind a new coordinate where 119910 = 0 is an invariant algebraiccurve of the transformed system (5)

We consider system (2) which we write into the form

= 119875 (119909 119910) = 119901119909 (1 + 1198750(119909)) + sum

119895ge1

119875119895(119909) 119910119895

= 119876 (119909 119910) = 1199091198760(119909) minus 119902119910 (1 + 119876

1(119909)) + sum

119895ge2

119876119895(119909) 119910119895

(19)

where 119875119895(119909) = sum

119894ge0119886119894119895119909119894 and 119876

119895(119909) = sum

119894ge0119887119894119895119909119894 are analytic

functions defined in a neighborhood of the origin

Abstract and Applied Analysis 5

Lemma 5 System (19) is orbitally equivalent to

= 119875 (119909 119910) = 119901119909 (1 + 1198750(119909)) + sum

119895ge1

119875119895(119909) 119910119895

= 119876 (119909 119910) = minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(20)

that is 1198760(119909) equiv 119876

1(119909) equiv 0 Moreover (119901 119902) are coprimes

Proof First we prove that we can achieve 1198760(119909) equiv 0 Assume

that 1198760(119909) equiv 0 and let 119873 = min119894 isin N cup 0 119887

1198940= 0 The

change of variable 119906 = 119909 V = 119910 minus (1198871198730

(119901(119873 + 1) + 119902))119909119873+1

transforms system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) V119895

V = 1199060(119906) minus 119902V (1 +

1(119906)) + sum

119895ge2

119895(119906) V119895

(21)

where 0(119906) = sum

119894ge01198871198940119906119894 and

1198940= 0 for 119894 = 0 1 119873

because V = minus ((119873 + 1)1198871198730

(119901(119873 + 1) + 119902))119906119873 hence

1199060(119906) = 119887

1198730119906119873+1

minus1199021198871198730

119901 (119873 + 1) + 119902119906119873+1

minus119901 (119873 + 1) 119887

1198730

119901 (119873 + 1) + 119902119906119873+1

+ O (119906119873+2

)

= O (119906119873+2

)

(22)

Therefore by means of successive change of variables we cantransform system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) 119910119895

V = minus119902V (1 + 1(119906)) + sum

119895ge2

119895(119906) V119895

(23)

To complete the proof it is enough to apply the scaling 119889119905 =

119889119879(1 + 1(119906))

Lemma 6 Let G be a formal function with G(0) = 0 Thesystem

(

) = (

119901119909

minus119902119910) + (

119901119909

119902119910)G (119909

119902119910119901) (24)

is formally integrable if and only ifG equiv 0

Proof The sufficient condition is trivial since 119909119902119910119901 is a first

integral of ( )119879 = (119901119909 minus119902119910)119879

We now will prove the necessary condition Let F(119909 119910) =(119901119909 minus119902119910)

119879+ (119901119909 119902119910)

119879G(ℎ) where ℎ = 119909

119902119910119901 and G(ℎ) =

sum119895ge1

119892119895ℎ119895 G equiv 0 otherwise the proof is finished We

consider 1198950= min119895 isin N | 119892

119895= 0 If 119868 is a first integral of

this system then there exists 119872 isin N such that 119868 = ℎ119872

+ sdot sdot sdot

and 0 = nabla119868 sdotF = 21199011199021198921198950ℎ1198950+1 + sdot sdot sdot Therefore 119892

1198950= 0 and this

is a contradiction

The next result shows that any integrable system (20)admits always a first integral of the form119867

1

Proposition 7 If system (20) is formally integrable then119867(119909 119910) = sum

119895ge0119891119895(119909)119910119895 with 119891

119895(119909) a formal function is a first

integral and moreover 1198910= sdot sdot sdot 119891

119901minus1= 0

Proof By applying the Poincare-Dulac normal form thereexists a change of variable to transform (20) into

(

V) = (

119901119906

minus119902V) + (

sum

119896ge1

119886119896119906ℎ119896

sum

119896ge1

119887119896Vℎ119896) (25)

where ℎ = 119906119902V119901 We can assume that the change of variable is

119906 = 119909 + sdot sdot sdot V = 119910(1 + sdot sdot sdot) because the axis 119910 = 0 and V = 0

are invariantConsidering the formal functions F(ℎ) = sum

119896ge1119888119896ℎ119896 and

G(ℎ) = sum119896ge1

119889119896ℎ119896 with 119888

119896= (119902119886119896minus 119901119887119896)2119901119902 and 119889

119896= (119902119886119896+

119901119887119896)2119901119902 we get

(

V) = (

119901119906

minus119902V) (1 +F (119906

119902V119901)) + (

119901119906

119902V) G (119906

119902V119901) (26)

whereF G are formal functions withF(0) = G(0) = 0Moreover by the scaling of the time 119889119905 = (1 +F)119889120591 we

can getF equiv 0 that is it is possible to transform (20) into

(1199061015840

V1015840) = (

119901119906

minus119902V) + (

119901119906

119902V)G (119906

119902V119901) (27)

whereG = G(1 +F) is a formal function withG(0) = 0If this system is formally integrable applying Lemma 6

we obtainG = 0Therefore119867(119909 119910) = (119909+sdot sdot sdot )119902119910119901(1+sdot sdot sdot )

119901=

sum119896ge119901

119891119896(119909)119910119896 is a first integral of system (20)

The main result of this work is as follows

Theorem 8 System (20) admits an analytic first integral119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896 if for each 119896 isin N such that 119896 = 119872119901

with119872 isin N is verified that the derivative (119872119902)119896

(0) = 0 where119896for 119896 ge 119901 is the analytical function

119896(119909) =

119892119896(119909)

119901 (1 + 1198750(119909)) 119891

119901(119909)119896

119891119901(119909) = exp(minusint

119909

0

(1199021198750(119904)

119901119904 (1 + 1198750(119904))

) 119889119904)

119892119901= 0 and for 119896 gt 119901

119892119896(119909) =

119896minus119901

sum

119895=1

((119896 minus 119895) 119891119896minus119895

(119909)119876119895+1

(119909) + 1198911015840

119896minus119895(119909) 119875119895(119909))

(28)

and for 119896 ge 119901

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

119904minus1minus(119896119902119901)

119896(119904) 119889119904) (29)

6 Abstract and Applied Analysis

Proof By Proposition 7 we have 1198910

= sdot sdot sdot = 119891119901minus1

= 0 Ifwe impose that 119867(119909 119910) be a first integral of system (20) weobtain that the first condition is

minus119901119902119891119901(119909) + 119901119909119891

1015840

119901(119909) (1 + 119875

0(119909)) = 0 (30)

The next coefficients for each power of 119910 are of the form

minus119902119896119891119896(119909) + 119901119909 (1 + 119875

0(119909)) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (31)

where 119892119901= 0 and 119892

119896(119909) for 119896 gt 119901 depends of the previous

119891119894(119909) for 119894 = 119901 119901 + 1 119896 minus 1 More specifically we obtain

the expression (28)The solution of homogeneous equation associated to (31)

is

119891ℎ

119896(119909) = exp(int

119896119902

119901119909 (1 + 1198750(119909))

) (32)

We can define the following analytical function 119887(119909) =

minus(119902119901119909)(1198750(119909)(1+119875

0(119909))) and the integrand of the previous

expression of 119891ℎ

119896is a rational function which admits the

following fraction decomposition

119896119902

119901119909 (1 + 1198750(119909))

=119896119902

119901119909+ 119896119887 (119909) (33)

Therefore 119891ℎ

119896(119909) = 119909

119896119902119901 exp(int 119896119887(119909)) =

119909119896119902119901

(exp(int1199090119887(119904)119889119904))

119896

and for the first equation whichcorresponds to 119896 = 119901 we have that 119892

119901(119909) = 0 Therefore

119891ℎ

119901(119909) has the form 119891

ℎ

119901(119909) = 119862

119901119909119902(119891119901(119909))119901

where 119891119901(119909) =

exp(int1199090119887(119904)119889119904) is an analytic function with 119891

119901(0) = 1 So

119891ℎ

119896(119909) = 119862

119896119909119896119902119901

119891119901(119909)119896 and the solution of (31) is given by

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

0

119904minus119896119902119901

119891119901(119904)minus119896

119901119904 (1 + 1198750(119904))

119892119896(119904) 119889119904)

(34)

Taking into account the form of 119892119896(119909) we obtain the

expression (29) for119891119896 Now it is straightforward to see that the

integral does not give logarithmic terms in the case 119896119902119901 notin N

and in the case 119896 = 119872119901 because (119872119902)

119896(0) = 0 If we choose

119862119896

= 0 we have that each 119891119896(119909) for all 119896 is an analytic

function and therefore system (20) under the assumptionsof the theorem admits a formal first integral of the form119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896

Corollary 9 System (20) admits a formal first integral of theform119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895 if one of the following conditions

holds

(a) 1198750(119909) equiv 0 and for each 119896 = 119872119901 119872 isin N 119892

119896(119909) is a

polynomial of degree at most119872119902 minus 1(b) 1198750(119909) equiv 119886119909

119903 with 119903 a positive integer and for each119896 = 119872119901 119872 isin N 119892

119896(119909) is a rational function of the

form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119902119903119901 where

119896is a

polynomial of degree at most119872119902 minus 1

Proof In case (a) it is enough to applyTheorem 8 for119891119901(119909) equiv

1 and 119896(119909) = 119892

119896(119909)119901 For 119896 = 119872119901 119872 isin N and we have

(119872119902)

119896(119909) = 119892

(119872119902)

119896119901 = 0 because the degree of 119892

119896(119909) is less

than 119902119872In case (b) we have that 119891

119901(119909) equiv (1 + 119886119909

119903)minus119902119903119901 and

119896(119909) = 119892

119896(119909)119901(1 + 119886119909

119903)1minus119902119903119901 Therefore

119896(119909) =

119896(119909)119901

and (119872119901)

119896(119909) = 0 because the degree of

119896(119909) is less than 119902119872

We obtain the result by applying also Theorem 8

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separate formin order to be applied directly to different examples

Lemma 10 If system (20)with 1198750(119909) equiv 0 admits a formal first

integral of the form119867(119909 119910) = sum119895ge119901

119891119895(119909)119910119895 then the following

conditions hold

(a) If for each 119896 such that 119896119902119901 notin N 119892119896is a polyno-

mial then it is possible to choose 119891119896polynomial with

deg(119891119896) = deg(119892

119896)

(b) If for each 119896 = 119872119901 119872 isin N 119892119896(119909) is a polynomial

with deg(119892119896) le 119872119902 minus 1 then it is possible to choose 119891

119896

polynomial with deg(119891119896) le 119872119902 minus 1

Proof By applying Theorem 8 we obtain 119891119901(119909) equiv 1 and

119896(119909) = 119892

119896(119909)119901 If we choose 119862

119896= 0 in (29) we obtain

119891119896(119909) = minus

1

119901119909119896119902119901

int

119909

119904minus1minus119896119902119901

119892119896(119904) 119889119904 (35)

In case (a) the result is trivial since the integral does not givelogarithmic terms because 119896119902119901 notin N

In case (b) 119896 = 119872119901 and 119891119896(119909) =

minus(1119901)119909119872119902

int119909

119904minus1minus119872119902

119892119896(119904) 119889119904Therefore we obtain the result

applying that 119892119896is a polynomial with deg(119892

119896) le 119872119902 minus 1

Based on the results presented in this work the followingproposition gives a large family of analytic systems that havean analytic first integral

Proposition 11 The analytic system

= 119901119909 + sum

119895ge1

119875119895(119909) 119910119895

= minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(36)

where 119875119895and 119876

119895are polynomials such that deg(119875

119895) le lfloor119895119902119901rfloor

deg(119876119895) le lfloor(119895 minus 1)119902119901rfloor minus 1 has an analytic first integral in

a neighborhood of the origin where lfloor119886rfloor is the integer part of119886 isin R

Proof From Proposition 7 if system (36) is integrable thena first integral is of the form 119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895

where 119891119895(119909) is a formal function Moreover 119875

0(119909) equiv 0 and

therefore by applying Theorem 8 we obtain 119891119901(119909) equiv 1

119896(119909) = 119892

119896(119909)119901 119892

119901= 0 and 119891

119901(119909) = 119909

119902

Abstract and Applied Analysis 7

We will prove now by induction that deg(119891119895) le lfloor119895119902119901rfloor

and deg(119892119895) le lfloor119895119902119901rfloor minus 1 for all 119895 isin N and 119901 le 119895

For 119895 = 119901 is true since 119891119901(119909) = 119909

119902 and 119892119901

= 0 If wesuppose that the hypothesis is true for 1 le 119895 le 119895

0minus1 we have

to prove that deg(1198921198950) le lfloor119895

0119902119901rfloor minus 1 and deg(119891

1198950) le lfloor119895

0119902119901rfloor

On the other hand applying (28) we have

deg (1198921198950)

le max1le119895le1198950minus119901

deg (1198911198950minus119895

)+deg (119876119895+1

) deg (11989110158401198950minus119895

) + deg (119875119895)

le lfloor1198950119902

119901rfloor minus 1

(37)

since for 1 le 119895 le 1198950minus 119901 we have

deg (1198911198950minus119895

) + deg (119876119895+1

) le lfloor(1198950minus 119895) 119902119901rfloor + lfloor

119895119902

119901rfloor minus 1

le lfloor1198950119902

119901rfloor minus 1

deg (11989110158401198950minus119895

) + deg (119875119895) le lfloor

(1198950minus 119895) 119902

119901rfloor minus 1 + lfloor

119895119902

119901rfloor

le lfloor1198950119902

119901rfloor minus 1

(38)

By applying Lemma 10 we have that deg(1198911198950) le lfloor119895

0119902119901rfloorminus

1 In particular deg(1198911198950) le lfloor119895

0119902119901rfloor

Therefore if 119896 = 119872119901 deg(119892119896) le 119872119902 minus 1 119892119872119902

119896(0) = 0 and

119872119902

119896(0) = 0To finish the proof it is enough to apply Theorem 8

However as it is difficult to determine the new coordi-nates where system (19) has 119876

0(119909) equiv 119876

1(119909) equiv 0 we are

going to work with the original system (5) imposing directlythese two conditions In order that one of the series 119867

1or

1198672be a formal first integral of system (5) we have that these

series must have the term 119909119902119910119901 as a first monomial in its

development We will see that in this case and under someconditionswe can use the inductionmethod in order to verifythe existence of a formal first integral

The following result is only established for the series 1198671

but we can obtain a similar result for the case where 1198672is a

formal first integral of system (5)

Proposition 12 We consider system (5)where wewrite1198651and

1198652into the form

1198651(119909 119910) =

infin

sum

119894+119895=2

119886119894119895119909119894119910119895

1198652(119909 119910) =

infin

sum

119894+119895=2

119887119894119895119909119894119910119895

(39)

where 119886119894119895and 119887119894119895are arbitrary constants If we impose that 119867

1

be a first integral of system (5) we obtain that the first conditionis

infin

sum

119894=2

1198871198940

1199091198941198911(119909) + 119901119909119891

1015840

0(119909) +

infin

sum

119894=2

11988611989401199091198941198911015840

0(119909) = 0 (40)

The proof of this proposition is also straightforwarddeveloping in powers of 119910 and taking the coefficient of thepower 119910

0 Hence in order to have a recursive differentialequations where each 119891

119894(119909) does not depend on 119891

119894+1(119909) we

must impose that 1198871198940

= 0 for all 119894 and consequently 1198910must

be a constant that we can take without loss of generality equalto zero The consequence as before is that if all 119887

1198940= 0 for all

119894 the system (5) has 119910 as invariant algebraic curveThe next coefficients for each power of 119910 are of the form

119896(minus119902 +

infin

sum

119894=1

1198871198941119909119894)119891119896(119909)

+ (119901119909 +

infin

sum

119894=2

1198861198940119909119894)1198911015840

119896(119909) + 119892

119896(119909) = 0

(41)

where 119892119896(119909) depends of the previous119891

119894(119909) for 119894 = 1 2 119896minus

1 and for the first equation which corresponds to 119896 = 1 wehave that 119892

1(119909) = 0 Therefore 119891

1(119909) has the form

1198911(119909) = exp(minusint

minus119902 + suminfin

119894=11198871198941119909119894

119901119909 + suminfin

119894=21198861198940119909119894

) (42)

The integrand of the previous expression of 1198911is a rational

function which admits the following fraction decomposition

minus119902 + suminfin

119894=11198871198941

119909119894

119901119909 + suminfin

119894=21198861198940

119909119894=

119860

119909+

119861 (119909)

119901 + suminfin

119894=21198861198940

119909119894minus1 (43)

where 119860 = minus119902119901 and 119861(119909) is a formal series Therefore1198911(119909) = 119909

1199021199011198911(119909) where 119891

1is the corresponding formal

series obtained after the integration Now we consider theparticular case 119901 = 1 The following theorem gives sufficientconditions to have formal integrability in this case A similarproposition can be established for the case when the firstintegral is of the form119867

2

Proposition 13 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if 119901 = 1

1198871198940= 0 for 119894 ge 2 and 119887

1198941= 0 for 119894 ge 1 and

119892119896(119909) =

119896(119909) 119875 (119909 0)

1199091198911(119909)minus119896

with (119896119902)

119896(0) = 0 (44)

Proof Under the assumptions of the theorem we have that1198910= 0 119891

1= 1199091199021198911(119909) and (41) takes the form

minus119896119902119891119896(119909) + 119875 (119909 0) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (45)

whose solution is given by

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909 119904minus119896119902

1198911(119904)minus119896

119875 (119904 0)119892119896(119904) 119889119904)

(46)

8 Abstract and Applied Analysis

Taking into account the form of 119892119896(119909) we obtain

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909

119904minus1minus119896119902

119896(119904) 119889119904) (47)

Now it is straightforward to see that the integral does not givelogarithmic terms because the 119896119902-th derivative of

119896(119904) at zero

is zero that is (119896119902)119896

(0) = 0 We have that each 119891119896(119909) for all

119896 is an analytic function and therefore system (5) under theassumptions of the theorem admits a formal first integral ofthe form (39)

Examples where Proposition 13 can be applied can befound in [7 12 16 17] where some partial results are givenWe also can establish the following corollary in the casewherethe 119892119896(119909) are polynomials or some irrational functions

Corollary 14 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if one of

the following conditions holds

(a) 119901 = 1 1198861198940

= 1198871198940

= 0 for 119894 ge 2 and 1198871198941

= 0 for 119894 ge 1 and119892119896(119909) is a polynomial of degree at most 119896119902 minus 1

(b) 119901 = 1 1198861198940= 0 for 119894 = 119903 + 1 119887

1198940= 0 for 119894 ge 2 and 119887

1198941= 0

for 119894 ge 1with 119903 a positive integer and 119892119896(119909) is a rational

function of the form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119896119902119903

where 119896is a polynomial of degree at most 119896119902 minus 1

Proof In case (a) for we have1198910= 0 and119891

1= 119909119902 (41) take the

form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119892

119896(119909) = 0 (48)

whose solution is given by

119891119896(119909) = 119862

119896119909119896119902

minus 119909119896119902

int

119909

119904minus1minus119896119902

119892119896(119904) 119889119904 (49)

and taking into account that 119892119896(119909) that depends on 119891

119894for 119894 =

1 2 119896 minus 1 is at most of degree 119896119902 minus 1 we obtain that 119891119896is

a polynomial Moreover in this case system (5) becomes

= 119901119909 + 1199101206011(119909 119910)

= minus119902119910 + 1199101206012(119909 119910)

(50)

where 1206011and 120601

2are analytic functions

In case (b) we have that 1198910= 0 119891

1= 119909119902(1 + 119886119909

119903)minus119902119903 and

(41) takes the form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119886119909

119903+11198911015840

119896(119909) + 119892

119896(119909) = 0 (51)

whose solution is given by

119891119896(119909) = 119909

119896119902(1 + 119886119909

119903)minus119896119902119903

times (119862119896minus int

119909

119904minus1minus119896119902

(1 + 119886119904)minus1+119896119902119903

119892119896(119904) 119889119904)

(52)

and taking into account the form of 119892119896(119909) we obtain that

119891119896(119909) is also of the form 119891

119896(119909) = 119901

119896(119909)(1 + 119886119909

119903)minus119896119902119904 where

119901119896(119909) is a polynomial because the integral in (52) does not

give logarithmic terms

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separateform in order to be applied directly to different examplesCorollary 14 generalizes a lot of cases obtained in the liter-ature when some concrete families of polynomials systemswere studiedWe have notmade an exhaustive study of all thecases included in these large families but for instance containsthe cases (3) and (4) of Theorem 3 in [17] and case (3) ofTheorem 2 in [16]

In resume when we study a family that satisfies thenecessary conditions we must impose if the family systemhas a formal first integral of the form 119867

1or 1198672 If none of

them work we can look for the proper coordinates (1199111 1199112)

defined by the stable and instable manifold of the 119901 minus119902

resonant singular point In this way we must study if at leastone of the separatrices of the resonant point is algebraic Thebest case is when both are algebraicTherefore the problem tofind proper coordinates (119911

1 1199112) becomes a problem of finding

the invariant algebraic curves of system (5) passing throughthe 119901 minus119902 resonant singular point The existence of invariantalgebraic curves takes a leading role in the formal integrabilitytheory and is the base to find the proper coordinates systemThis is shown in the examples provided in [12] that aresolved by ad hoc methods and where the changes of variablesproposed to solve the examples are always invariant algebraiccurves of the original system passing through the originIn fact in [12] it recalled the Abhyankar-Moch theorem[37] This theorem establishes assuming the existence of aninvariant rational algebraic curve passing through the originthe existence of a rational invertible change of variables suchthat the invariant curve becomes one of the invariant axesIn this paper this result is generalized in the sense that any119901 minus119902 resonant singular point is formal orbitally equivalentto a system with 119910 = 0 as invariant algebraic curve seeLemma 5

6 Examples

In this section we present some examples where the resultsdeveloped in this paper are applied

Example 15 We first consider the 1 minus1 resonant quadraticsystem given by

= 119909 + 119886201199092+ 11988611119909119910 + 119886

021199102

= minus119910 + 119887201199092+ 11988711119909119910 + 119887

021199102

(53)

The aim is not to do an exhaustive study of when system (53)has a formal first integral in a neighborhood of the originIn fact the complex center cases were studied by Dulac see[34] The aim is only to show how to use the results givenin this paper If we apply directly Corollary 14 statement (a)we obtain 119887

20= 0 (this condition implies that 119910 = 0 be an

algebraic invariant curve of system (53)) and 11988620

= 11988711

= 0 Inthis case the quadratic system (53) reads

= 119909 + 11988611119909119910 + 119886

021199102

= minus119910 + 119887021199102

(54)

Abstract and Applied Analysis 9

and admits a formal first integral and by Theorem 2 system(54) has an analytic first integral in a neighborhood of theorigin

In fact the associated equation to system (54) is a linearequation with respect to 119909 and the system has a Darboux firstintegral of the form

119867(119909 119910)

= (11988702119910 minus 1)

minus1minus(1198861111988702)

times [211988602

minus 2 (1198860211988611

+ 1198860211988702) 119910

+ (1198863

11minus 119886111198872

02) 119909119910 + (119886

021198862

11+ 119886021198861111988702) 1199102]

(55)

Example 16 Consider the 1 minus1 resonant LotkandashVolterraplanar complex quartic system given by

= 119909 (1 minus 11988610119909 minus 11988601119910 minus 119886201199092minus 11988611119909119910

minus119886021199102minus 119886301199093minus 119886211199092119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988710119909 minus 11988701119910 minus 119887201199092minus 11988711119909119910

minus119887021199102minus 119887301199093minus 119887211199092119910 minus 119887121199091199102minus 119887031199103)

(56)

The LotkandashVolterra planar complex systems have been stud-ied in several papers see [7 11 13ndash15] In any case the LotkandashVolterra planar complex quartic system is an open problemcomputationally infeasible However with the results of thispaperwe are going to see that we can give some sufficient con-ditions to have an analytic first integral without computingany resonant focus value We propose a formal first integralof the form 119867

1 that is of the form (16) The recurrence that

gives119891119894(119909) is given by minus119894119891

119894(119909)+119909119891

1015840

119894(119909)+119892

119894(119909) = 0where 119892

119894(119909)

is(119894 minus 3) 119887

03119891119894minus3

(119909) + (119894 minus 2) 11988702119891119894minus2

(119909) + (119894 minus 2) 11988712119909119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) + (119894 minus 1) 11988711119909119891119894minus1

(119909)

+ (119894 minus 1) 119887211199092119891119894minus1

(119909)

+ 11989411988710119909119891119894(119909) + 119894119887

201199092119891119894(119909)

+ 119894119887301199093119891119894(119909) minus 119886

031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909) minus 119886

1111990921198911015840

119894minus1(119909)

minus 1198861011990921198911015840

119894(119909) minus 119886

2011990931198911015840

119894(119909) minus 119886

3011990941198911015840

119894(119909)

(57)

Now we impose that 119892119894(119909) must be of degree lei minus1 This

implies that 11988610

= 11988620

= 11988630

= 11988710

= 11988720

= 11988730

= 0 and11988721

= 11988621

= 11988711

= 11988611

= 0 and in this case the recurrence isminus119894119891119894(119909) + 119909119891

1015840

119894(119909) + 119892

119894(119909) = 0 where 119892

119894(119909) is given by

(119894 minus 3) 11988703119891119894minus3

(119909) + (119894 minus 2) (11988702

+ 11988712119909)119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) minus 119886031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909)

(58)

We obtain that its solution is given by

119891119894(119909) = 119862

119894119909119894minus 119909119894int

119909

119904minus1minus119894

119892119894(119904) 119889119904 (59)

and taking into account that 119892119894(119909) is of degree le 119894 minus 1 we can

establish the following result

Proposition 17 The system

= 119909 (1 minus 11988601119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988701119910 minus 119887021199102minus 119887121199091199102minus 119887031199103)

(60)

has an analytic first integral in a neighborhood of the origin

In fact Proposition 17 is a consequence of Proposition 11This technique can be applied to several families of polyno-mial systems and in fact gives a new mechanism to obtainresonant centers and also real centers for the complex systemsthat have a real preimage (when this preimage exists see eg[18])

Acknowledgments

A Algaba and C Garcıa are supported by aMICINNFEDERGrant no MTM2010-20907-C02-02 and by the Consejerıade Educacıon y Ciencia de la Junta de Andalucıa (projectsEXC2008FQM-872 TIC-130 and FQM-276) J Gine is par-tially supported by a MICINNFEDER Grant no MTM2011-22877 and by Generalitat de Catalunya Grant no 2009SGR381

References

[1] AAlgaba E Gamero andCGarcıa ldquoThe integrability problemfor a class of planar systemsrdquo Nonlinearity vol 22 no 2 pp395ndash420 2009

[2] A Algaba E Freire E Gamero and C Garcıa ldquoMonodromycenter-focus and integrability problems for quasi-homogeneouspolynomial systemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1726ndash1736 2010

[3] A Algaba N Fuentes and C Garcıa ldquoCenters of quasi-homogeneous polynomial planar systemsrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 419ndash431 2012

[4] A Algaba C Garcıa and M Reyes ldquoExistence of an inverseintegrating factor center problem and integrability of a class ofnilpotent systemsrdquo Chaos Solitons amp Fractals vol 45 no 6 pp869ndash878 2012

[5] J Chavarriga H Giacomini J Gine and J Llibre ldquoOn theintegrability of two-dimensional flowsrdquo Journal of DifferentialEquations vol 157 no 1 pp 163ndash182 1999

[6] H Zoładek ldquoThe problem of center for resonant singular pointsof polynomial vector fieldsrdquo Journal of Differential Equationsvol 137 no 1 pp 94ndash118 1997

[7] X Chen J Gine V G Romanovski and D S Shafer ldquoThe1 -q resonant center problem for certain cubic Lotka-Volterrasystemsrdquo Applied Mathematics and Computation vol 218 no23 pp 11620ndash11633 2012

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

resonant centers obviating the computation of the necessaryconditions (see for instance Proposition 11)

2 Definitions and Preliminary Concepts

We first consider a real system of ordinary differentialequations onR2 with an isolated singular point at the originwhose linear parts are nonzero pure imaginary numbers Bya linear change of coordinates and a constant time rescalingthe system takes the form

= V + sdot sdot sdot V = minus119906 + sdot sdot sdot (1)

The classical Poincare-Liapunov center theorem states thatthe origin is a center if and only if the system admits ananalytic first integral of the form 120601(119906 V) = 119906

2+ V2 + sdot sdot sdot see

for instance [18 28ndash31] and the references thereinIf system (1) is complexified in a natural way taking

119911 = 119906 + 119894V then we obtain a differential equation of theform = 119894119911 + sdot sdot sdot In this case one constructs step bystep the formal first integral Φ = 119911119911 + sdot sdot sdot satisfying theequation Φ = V

3|119911|4+ V5|119911|6+ sdot sdot sdot where the coefficients

V2119894+1

called the focus quantities are polynomials in thecoefficients of the original system The theorem of Poincare-Liapunov [32 33] says that the point 119911 = 0 is a center ifand only if all the V

2119894+1= 0 Existence of a first integral 120601

is equivalent to existence of an analytic first integral for thecomplexified equation of the form Φ = 119911119911 + sdot sdot sdot Takingthe complex conjugated equation there arises an analyticsystem of ordinary differential equations on C2 of the form = 119894119911 + sdot sdot sdot = minus119894119908 + sdot sdot sdot where 119908 = 119906 minus 119894V Hence after thecomplexification the system is transformed into an analyticsystem with eigenvalues +119894 and minus119894 This is the 1 minus1 resonantsingular point and the numbers V

2119894+1become the coefficients

of the resonant terms in its orbital normal form Dulac [34]chosen this way to approach the center problem for quadraticsystems

The next natural generalization is to consider the case ofan analytic vector field inC2 with 119901 minus119902 resonant elementarysingular point

= 119901119909 + sdot sdot sdot = minus119902119910 + sdot sdot sdot (2)

where 119901 119902 isin Z with 119901 119902 gt 0 These facts motivate thegeneralization of the concept of real center to certain classesof systems of ordinary differential equations on C2 In thiscase we have the following definition of a resonant center orfocus coming from Dulac [34] see also [6]

Definition 1 A 119901 minus119902 resonant elementary singular point ofan analytic system is a resonant center if and only if thereexists a local meromorphic first integral Φ = ℎ

0+ sdot sdot sdot with

ℎ0= 119909119902119910119901 This singular point is a resonant focus of order 119896

if and only if there is a formal power series Φ = 119909119902119910119901+ sdot sdot sdot

with the property Φ = 119892119896ℎ119896+1

0+ sdot sdot sdot

Recently several works are focused on the study ofresonant centers for complex analytic systems In theseworksthe integrability and linearizability problems are studied

The linearizability problem is focused on the study of theexistence of an analytic change of coordinates that linearizethe complex analytic system

Mattei and Moussu [27] proved the next result for allisolated singularities

Theorem 2 Assume that system (1) or (2) with an isolatedsingularity at the origin has a formal first integral 119867 isin

R[[119909 119910]] around it Then there exists an analytic first integralaround the singularity

In the light of the former result we can conclude that inorder to prove the existence of an analytic first integral weonly need to prove the existence of a formal first integralTherefore the formal integrability problem takes a primaryrole for the upcoming investigations on the center andresonant center problem

3 Blowup of Resonant Saddles

In this section we will consider the first method to approachthe formal integrability problem for resonant centers Weconsider the resonant analytic system (2) We now do theblowup of this singularity This means that we apply the map(119909 119910) rarr (119909 119911) = (119909 119910119909) The point 119909 = 119910 = 0 is replacedby the line 119909 = 0 which contains two singular points thatcorrespond to the separatrices of (2) and are the saddles 119901

1

which is (119901 + 119902) minus119901 resonant and 1199012which is (119901 + 119902) minus119902

resonant The following lemma is proved in [12] using thenormal orbital form of a resonant analytic system (2)

Lemma 3 If one of the points 1199011or 1199012is orbitally analytically

linearizable then the point 119909 = 119910 = 0 is a 119901 minus119902 resonantcenter

As a corollary of this lemma we get the following usefulproperty which can be used to approach the resonant centerproblem If the point 119901

1is orbitally analytically linearizable

then it has a formal first integral in the variables (119909 119911) of theform

119867 = 119909119901119911119902+119901

infin

sum

119899=0

119891119899(119909) 119911119899 (3)

with 119891119899(0) = 0 Hence the original 119901 minus119902 resonant point has

a nonformal first integral of the form

119909minus119902119910119902+119901

infin

sum

119899=0

119891119899(119909) 119909minus119899119910119899=

infin

sum

119899=0

119891119899(119909) 119909minus119902minus119899

119910119902+119901+119899

(4)

Consequently if the resonant point 119901 minus119902 has a non formalfirst integral of the form (4) then by Lemma 3 the resonantpoint 119901 minus119902 is a resonant center The same result can beestablished for the other saddle119901

2These types of results were

also given by Bruno in [35 36]

Abstract and Applied Analysis 3

4 The 120576-Method for Resonant Centers

We recall in this section the 120576-method developed in [23]which we apply here to resonant centers We consider system(2) which we write into the form

= 119875 (119909 119910) = 119901119909 + 1198651(119909 119910)

= 119876 (119909 119910) = minus119902 119910 + 1198652(119909 119910)

(5)

where 1198651(119909 119910) and 119865

2(119909 119910) are analytic functions without

constant and linear terms defined in a neighborhood of theorigin

To implement the algorithm we introduce a rescalingof the variables and a time rescaling given by (119909 119910 119905) rarr

(120576119901119909 120576119902119910 120576119903119905) where 120576 gt 0 and 119901 119902 and 119903 isin Z and system

(5) takes the form

= 120576119903minus119901

(119901 120576119901119909 + 1198651(120576119901119909 120576119902119910))

= 120576119903minus119902

(minus119902 120576119902119910 + 1198652(120576119901119909 120576119902119910))

(6)

We choose 119901 119902 119903 in such a way that system (6) will beanalytic in 120576 Hence by the classical theorem of the analyticdependence with respect to the parameters we have thatsystem (6) admits a first integral which can be developedin power of series of 120576 because it is analytic with respectto this parameter Therefore we can propose the followingdevelopment for the first integral

119867(119909 119910) =

infin

sum

119896=0

120576119896ℎ119896(119909 119910) (7)

where ℎ119896(119909 119910) are arbitrary functionsWenotice that119865

1(119909 119910)

and 1198652(119909 119910) are analytic functions where both can be null

and 1198651(0 0) = 119865

2(0 0) = 0 so we can develop them in a

neighborhood of the origin as convergent series of 119909 and 119910

of the form

1198651(119909 119910) = 119901

119899(119909 119910) + 119901

119899+1(119909 119910) + sdot sdot sdot + 119901

119895(119909 119910) + sdot sdot sdot

1198652(119909 119910) = 119902

119899(119909 119910) + 119902

119899+1(119909 119910) + sdot sdot sdot + 119902

119895(119909 119910) + sdot sdot sdot

(8)

with 119899 = minsubdeg|(00)

1198651(119909 119910) subdeg

|(00)1198652(119909 119910) ge 2

We recall that given an analytic function 119891(119909 119910)

defined in a neighborhood of a point (1199090 1199100) we define

subdeg|(1199090 1199100)

119891(119909 119910) as the least positive integer 119895 such thatsome derivative (120597119895119891120597119909119894120597119910119895minus119894)(119909

0 1199100) is not zero We notice

that this computation depends on the variables (119909 119910) onwhich the function 119891(119909 119910) depends so we will explicit thevariables used in each computation of subdeg For instancesubdeg

|(1199090 1199100)119891(119909 119910) = 0 if and only if 119891(119909

0 1199100) = 0 In (8)

119901119895(119909 119910) and 119902

119895(119909 119910) denote homogeneous polynomials of 119909

and 119910 of degree 119895 ge 119899 It is possible that 119901119899(119909 119910) or 119902

119899(119909 119910)

be null but by definition not both of them can be null Thesimplest case is to consider in the rescaling 119901 = 119902 = 1 Infact this case is equivalent to impose that system (8) has afirst integral which can be expanded as a formal series inhomogeneous parts

The richness of the 120576-method is that using the parameter120576 the functions ℎ

119896(119909 119910) need not be homogeneous parts

and we can construct also a singular series expansion in thevariables 119909 and 119910 see [23] The method depends heavily onℎ0the first integral of the initial quasi-homogeneous system

The simpler is ℎ0farther we go with the method However ℎ

0

can be chosen using different scalings of variables (119909 119910 119905) rarr

(120576119901119909 120576119902119910 120576119903119905) where 120576 gt 0 and 119901 119902 and 119903 isin Z in such

a way that ℎ0will be as simple as possible The method

gives necessary conditions to have analytic integrability ora singular series expansion around a singular point andinformation about what is called in [23] the essential variablesof a system In the method developed in [23] the parameter 120576needs not be small The parameter 120576 may be relatively large(for instance 120576 rarr 1) The convergence of series (7) withrespect to (119909 119910) must be analyzed in each particular caseand the convergent rate depends upon the nonlinear termsof the system (8) In the case of a resonant center the mostconvenient ℎ

0is 119909119902119910119901

From now on we suppose that system (5) admits aformal first integral Therefore ℎ

119896(119909 119910) are homogeneous

polynomials of degree 119896 and we can take ℎ0(119909 119910) = 0 In this

case if we impose that series (7) be a formal first integral ofsystem (5) we obtain at each order of homogeneity

119901119909120597ℎ119896

120597119909(119909 119910) minus 119902119910

120597ℎ119896

120597119910(119909 119910) + 119866

119896(119909 119910) = 0 (9)

where119866119896(119909 119910) depends on the previous ℎ

119894for 119894 = 1 119896minus1

The solution of the partial equation (9) is

ℎ119896(119909 119910) = 119865 (119909

119902119901119910) minus int

119909 119866119896(119904 (119909119902119901

119910) 119904119902119901

)

119901 119904119889119904 (10)

where 119865 is an arbitrary function of 119909119902119901119910 We require that ℎ119896

be a polynomial and therefore an ℎ119896not having logarithmic

terms In order to avoid the logarithmic terms it is easy to seethat the function119866

119896(119909 119910)must not have polynomial terms of

the form (119909119902119910119901)119898 for 119898 isin N This result is the same as that

we obtain using normal form theory see [6]We must impose the vanishing of all the logarithmic

terms (if there exists any difference from zero) and prove thatall the functions ℎ

119894(119909 119910) for 119894 gt 1 are polynomials In fact

we must prove this by induction assuming that all the ℎ119894for

119894 = 1 119896 minus 1 are polynomials and prove that the recursivepartial differential equation with respect to ℎ

119896(see (9)) gives

also a polynomial This is not so easy and even less workingwith a partial recursive differential equation as (9)

At this point we think in the blow-up 119911 = 119910119909 becauseeach ℎ

119894(119909 119910) are homogeneous polynomials of degree 119894

4 Abstract and Applied Analysis

This blowup transforms the system (5) into a system ofvariables (119909 119911) of the form

= minus (119901 + 119902) 119911 + 119909F (119909 119911)

= 119901119909 + 1199092G (119909 119911)

(11)

where F(0 0) = 0 After that we propose a formal firstintegral of the form

=

infin

sum

119894ge1

119891119894(119911) 119909119894 (12)

where 119891119894(119911) must be polynomials of degree lei (if the log-

arithmic terms are zero) If we impose that series (12) bea first integral of the system in variables (119909 119911) we obtainthat the term 119887

201199092 must be zero However we can vanish

this term doing the change 119910 = 119910 minus 1205721199092 and 119909 = 119909 with

120572 = minus11988720(2119901 + 119902) We must also take 119891

1(119911) = 0 Finally at

each power of 119910 we have the recursive equation

119896119901119891119896(119911) minus (119901 + 119902) 119911119891

1015840

119896(119911) + 119892

119896(119911) = 0 (13)

where 119892119896(119911) depends of the previous functions 119891

119894for 119894 =

1 119896 minus 1 Hence in this case instead of getting the partialrecursive differential equation (9) we obtain an ordinaryrecursive differential equation whose solution is given by

119891119896(119911) =

119896119911119896119901(119901+119902)

+ 119911119896119901(119901+119902)

int

119911 119904minus1minus119896119901(119901+119902)

119901 + 119902119892119896(119904) 119889119904

(14)

where 119896is an arbitrary constant However we are going to

see that although we place in a resonant center where all the119891119896be polynomials this blowup does not allow us to prove in

general by induction of the existence of a formal first integralof the form (12)

In this sense consider the recursive differential equation(13) whose solution is (14) and we assume that system (5) hasa resonant degenerate center at the origin It is easy to see thatthere always exists a value 119896

0such that for 119896 ge 119896

0the arbitrary

polynomials 119891119894(119911) for 119894 = 1 119896

0minus 1 can gives following the

recursive equation (13) logarithmic terms Therefore we cannot apply the induction method to prove that the recursiveequation (13) gives always a polynomial

To see this we must to see that the solution (14) of therecursive equation (13) can give a logarithmic term Thishappens when

minus1 minus1198960119901

119901 + 119902+ 119898119896= minus1 (15)

where119898119896is the degree of the polynomial 119892

119896(119904) From here we

have that 1198960= 119898119896(119901+119902)119901 Hence if119901 = 1 then 119896

0= 119898119896(1+119902)

which can be satisfied because119898119896and 119902 are positive integers

For the case 119901 = 1 and taking into account that the value of119898119896increases when 119896 increases it can also exist a value of119898

119896

such that 119898119896is divisible by 119901 and that gives the value of 119896

0

that can give logarithmic terms

Therefore the conclusion is that the formal constructionof the first integral (7) using homogeneous terms or using theblow-up 119911 = 119910119909 and the formal series (12) do not allow touse the induction method in order to verify the existence of aformal first integral We must use other developments whichis the subject of the next section

5 Other Developments of the FormalFirst Integral

In this section we consider other developments of the formalfirst integral We consider the formal development of the firstintegral of system (5) in a series in the variable 119909 or in 119910 thatis we consider

1198671=

infin

sum

119896=0

119891119896(119909) 119910119896 or 119867

2=

infin

sum

119896=0

119892119896(119910) 119909119896 (16)

First we consider a general analytic system that we canalways write into the following forms

= 119891 (119909) + 119910Φ1(119909 119910) = 119892 (119909) + 119910Φ

2(119909 119910) (17)

or

= 119891 (119910) + 119909Ψ1(119909 119910) = 119892 (119910) + 119909Ψ

2(119909 119910) (18)

where 119891 119892Φ1Φ2 Ψ1 and Ψ

2are analytic in their variables

For systems (17) and (18) we have the following straightfor-ward result

Proposition 4 If we impose that the series 1198671(1198672 resp) be

a first integral of system (17) (system (18) resp) among othersthe first condition is 1198911015840

0(119909)119891(119909) + 119891

1(119909)119892(119909) = 0 (1198911015840

0(119910)119892(119910) +

1198911(119910)119891(119910) = 0 resp)

In order that this condition generates a collection ofrecursive differential equations where each 119891

119894(119909) does not

depend on119891119894+1

(119909) wemust impose119892(119909) = 0 (119891(119910) = 0 resp)which implies that119910 = 0 (119909 = 0 resp) is an invariant algebraiccurve of system (17) (system (18) resp)

In fact for system (5) there are always a new coordinates(1199111 1199112) where 119911

1= 0 and 119911

2= 0 are invariant curves These

invariant curves are defined by the stable and instable man-ifold of the p minusq resonant singular point and therefore theprevious conditions are directly satisfied In these coordinatesany 119901 minus119902 resonant singular point is a Lotka-Volterra systemIn the following result we are going to see that we always canfind a new coordinate where 119910 = 0 is an invariant algebraiccurve of the transformed system (5)

We consider system (2) which we write into the form

= 119875 (119909 119910) = 119901119909 (1 + 1198750(119909)) + sum

119895ge1

119875119895(119909) 119910119895

= 119876 (119909 119910) = 1199091198760(119909) minus 119902119910 (1 + 119876

1(119909)) + sum

119895ge2

119876119895(119909) 119910119895

(19)

where 119875119895(119909) = sum

119894ge0119886119894119895119909119894 and 119876

119895(119909) = sum

119894ge0119887119894119895119909119894 are analytic

functions defined in a neighborhood of the origin

Abstract and Applied Analysis 5

Lemma 5 System (19) is orbitally equivalent to

= 119875 (119909 119910) = 119901119909 (1 + 1198750(119909)) + sum

119895ge1

119875119895(119909) 119910119895

= 119876 (119909 119910) = minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(20)

that is 1198760(119909) equiv 119876

1(119909) equiv 0 Moreover (119901 119902) are coprimes

Proof First we prove that we can achieve 1198760(119909) equiv 0 Assume

that 1198760(119909) equiv 0 and let 119873 = min119894 isin N cup 0 119887

1198940= 0 The

change of variable 119906 = 119909 V = 119910 minus (1198871198730

(119901(119873 + 1) + 119902))119909119873+1

transforms system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) V119895

V = 1199060(119906) minus 119902V (1 +

1(119906)) + sum

119895ge2

119895(119906) V119895

(21)

where 0(119906) = sum

119894ge01198871198940119906119894 and

1198940= 0 for 119894 = 0 1 119873

because V = minus ((119873 + 1)1198871198730

(119901(119873 + 1) + 119902))119906119873 hence

1199060(119906) = 119887

1198730119906119873+1

minus1199021198871198730

119901 (119873 + 1) + 119902119906119873+1

minus119901 (119873 + 1) 119887

1198730

119901 (119873 + 1) + 119902119906119873+1

+ O (119906119873+2

)

= O (119906119873+2

)

(22)

Therefore by means of successive change of variables we cantransform system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) 119910119895

V = minus119902V (1 + 1(119906)) + sum

119895ge2

119895(119906) V119895

(23)

To complete the proof it is enough to apply the scaling 119889119905 =

119889119879(1 + 1(119906))

Lemma 6 Let G be a formal function with G(0) = 0 Thesystem

(

) = (

119901119909

minus119902119910) + (

119901119909

119902119910)G (119909

119902119910119901) (24)

is formally integrable if and only ifG equiv 0

Proof The sufficient condition is trivial since 119909119902119910119901 is a first

integral of ( )119879 = (119901119909 minus119902119910)119879

We now will prove the necessary condition Let F(119909 119910) =(119901119909 minus119902119910)

119879+ (119901119909 119902119910)

119879G(ℎ) where ℎ = 119909

119902119910119901 and G(ℎ) =

sum119895ge1

119892119895ℎ119895 G equiv 0 otherwise the proof is finished We

consider 1198950= min119895 isin N | 119892

119895= 0 If 119868 is a first integral of

this system then there exists 119872 isin N such that 119868 = ℎ119872

+ sdot sdot sdot

and 0 = nabla119868 sdotF = 21199011199021198921198950ℎ1198950+1 + sdot sdot sdot Therefore 119892

1198950= 0 and this

is a contradiction

The next result shows that any integrable system (20)admits always a first integral of the form119867

1

Proposition 7 If system (20) is formally integrable then119867(119909 119910) = sum

119895ge0119891119895(119909)119910119895 with 119891

119895(119909) a formal function is a first

integral and moreover 1198910= sdot sdot sdot 119891

119901minus1= 0

Proof By applying the Poincare-Dulac normal form thereexists a change of variable to transform (20) into

(

V) = (

119901119906

minus119902V) + (

sum

119896ge1

119886119896119906ℎ119896

sum

119896ge1

119887119896Vℎ119896) (25)

where ℎ = 119906119902V119901 We can assume that the change of variable is

119906 = 119909 + sdot sdot sdot V = 119910(1 + sdot sdot sdot) because the axis 119910 = 0 and V = 0

are invariantConsidering the formal functions F(ℎ) = sum

119896ge1119888119896ℎ119896 and

G(ℎ) = sum119896ge1

119889119896ℎ119896 with 119888

119896= (119902119886119896minus 119901119887119896)2119901119902 and 119889

119896= (119902119886119896+

119901119887119896)2119901119902 we get

(

V) = (

119901119906

minus119902V) (1 +F (119906

119902V119901)) + (

119901119906

119902V) G (119906

119902V119901) (26)

whereF G are formal functions withF(0) = G(0) = 0Moreover by the scaling of the time 119889119905 = (1 +F)119889120591 we

can getF equiv 0 that is it is possible to transform (20) into

(1199061015840

V1015840) = (

119901119906

minus119902V) + (

119901119906

119902V)G (119906

119902V119901) (27)

whereG = G(1 +F) is a formal function withG(0) = 0If this system is formally integrable applying Lemma 6

we obtainG = 0Therefore119867(119909 119910) = (119909+sdot sdot sdot )119902119910119901(1+sdot sdot sdot )

119901=

sum119896ge119901

119891119896(119909)119910119896 is a first integral of system (20)

The main result of this work is as follows

Theorem 8 System (20) admits an analytic first integral119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896 if for each 119896 isin N such that 119896 = 119872119901

with119872 isin N is verified that the derivative (119872119902)119896

(0) = 0 where119896for 119896 ge 119901 is the analytical function

119896(119909) =

119892119896(119909)

119901 (1 + 1198750(119909)) 119891

119901(119909)119896

119891119901(119909) = exp(minusint

119909

0

(1199021198750(119904)

119901119904 (1 + 1198750(119904))

) 119889119904)

119892119901= 0 and for 119896 gt 119901

119892119896(119909) =

119896minus119901

sum

119895=1

((119896 minus 119895) 119891119896minus119895

(119909)119876119895+1

(119909) + 1198911015840

119896minus119895(119909) 119875119895(119909))

(28)

and for 119896 ge 119901

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

119904minus1minus(119896119902119901)

119896(119904) 119889119904) (29)

6 Abstract and Applied Analysis

Proof By Proposition 7 we have 1198910

= sdot sdot sdot = 119891119901minus1

= 0 Ifwe impose that 119867(119909 119910) be a first integral of system (20) weobtain that the first condition is

minus119901119902119891119901(119909) + 119901119909119891

1015840

119901(119909) (1 + 119875

0(119909)) = 0 (30)

The next coefficients for each power of 119910 are of the form

minus119902119896119891119896(119909) + 119901119909 (1 + 119875

0(119909)) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (31)

where 119892119901= 0 and 119892

119896(119909) for 119896 gt 119901 depends of the previous

119891119894(119909) for 119894 = 119901 119901 + 1 119896 minus 1 More specifically we obtain

the expression (28)The solution of homogeneous equation associated to (31)

is

119891ℎ

119896(119909) = exp(int

119896119902

119901119909 (1 + 1198750(119909))

) (32)

We can define the following analytical function 119887(119909) =

minus(119902119901119909)(1198750(119909)(1+119875

0(119909))) and the integrand of the previous

expression of 119891ℎ

119896is a rational function which admits the

following fraction decomposition

119896119902

119901119909 (1 + 1198750(119909))

=119896119902

119901119909+ 119896119887 (119909) (33)

Therefore 119891ℎ

119896(119909) = 119909

119896119902119901 exp(int 119896119887(119909)) =

119909119896119902119901

(exp(int1199090119887(119904)119889119904))

119896

and for the first equation whichcorresponds to 119896 = 119901 we have that 119892

119901(119909) = 0 Therefore

119891ℎ

119901(119909) has the form 119891

ℎ

119901(119909) = 119862

119901119909119902(119891119901(119909))119901

where 119891119901(119909) =

exp(int1199090119887(119904)119889119904) is an analytic function with 119891

119901(0) = 1 So

119891ℎ

119896(119909) = 119862

119896119909119896119902119901

119891119901(119909)119896 and the solution of (31) is given by

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

0

119904minus119896119902119901

119891119901(119904)minus119896

119901119904 (1 + 1198750(119904))

119892119896(119904) 119889119904)

(34)

Taking into account the form of 119892119896(119909) we obtain the

expression (29) for119891119896 Now it is straightforward to see that the

integral does not give logarithmic terms in the case 119896119902119901 notin N

and in the case 119896 = 119872119901 because (119872119902)

119896(0) = 0 If we choose

119862119896

= 0 we have that each 119891119896(119909) for all 119896 is an analytic

function and therefore system (20) under the assumptionsof the theorem admits a formal first integral of the form119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896

Corollary 9 System (20) admits a formal first integral of theform119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895 if one of the following conditions

holds

(a) 1198750(119909) equiv 0 and for each 119896 = 119872119901 119872 isin N 119892

119896(119909) is a

polynomial of degree at most119872119902 minus 1(b) 1198750(119909) equiv 119886119909

119903 with 119903 a positive integer and for each119896 = 119872119901 119872 isin N 119892

119896(119909) is a rational function of the

form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119902119903119901 where

119896is a

polynomial of degree at most119872119902 minus 1

Proof In case (a) it is enough to applyTheorem 8 for119891119901(119909) equiv

1 and 119896(119909) = 119892

119896(119909)119901 For 119896 = 119872119901 119872 isin N and we have

(119872119902)

119896(119909) = 119892

(119872119902)

119896119901 = 0 because the degree of 119892

119896(119909) is less

than 119902119872In case (b) we have that 119891

119901(119909) equiv (1 + 119886119909

119903)minus119902119903119901 and

119896(119909) = 119892

119896(119909)119901(1 + 119886119909

119903)1minus119902119903119901 Therefore

119896(119909) =

119896(119909)119901

and (119872119901)

119896(119909) = 0 because the degree of

119896(119909) is less than 119902119872

We obtain the result by applying also Theorem 8

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separate formin order to be applied directly to different examples

Lemma 10 If system (20)with 1198750(119909) equiv 0 admits a formal first

integral of the form119867(119909 119910) = sum119895ge119901

119891119895(119909)119910119895 then the following

conditions hold

(a) If for each 119896 such that 119896119902119901 notin N 119892119896is a polyno-

mial then it is possible to choose 119891119896polynomial with

deg(119891119896) = deg(119892

119896)

(b) If for each 119896 = 119872119901 119872 isin N 119892119896(119909) is a polynomial

with deg(119892119896) le 119872119902 minus 1 then it is possible to choose 119891

119896

polynomial with deg(119891119896) le 119872119902 minus 1

Proof By applying Theorem 8 we obtain 119891119901(119909) equiv 1 and

119896(119909) = 119892

119896(119909)119901 If we choose 119862

119896= 0 in (29) we obtain

119891119896(119909) = minus

1

119901119909119896119902119901

int

119909

119904minus1minus119896119902119901

119892119896(119904) 119889119904 (35)

In case (a) the result is trivial since the integral does not givelogarithmic terms because 119896119902119901 notin N

In case (b) 119896 = 119872119901 and 119891119896(119909) =

minus(1119901)119909119872119902

int119909

119904minus1minus119872119902

119892119896(119904) 119889119904Therefore we obtain the result

applying that 119892119896is a polynomial with deg(119892

119896) le 119872119902 minus 1

Based on the results presented in this work the followingproposition gives a large family of analytic systems that havean analytic first integral

Proposition 11 The analytic system

= 119901119909 + sum

119895ge1

119875119895(119909) 119910119895

= minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(36)

where 119875119895and 119876

119895are polynomials such that deg(119875

119895) le lfloor119895119902119901rfloor

deg(119876119895) le lfloor(119895 minus 1)119902119901rfloor minus 1 has an analytic first integral in

a neighborhood of the origin where lfloor119886rfloor is the integer part of119886 isin R

Proof From Proposition 7 if system (36) is integrable thena first integral is of the form 119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895

where 119891119895(119909) is a formal function Moreover 119875

0(119909) equiv 0 and

therefore by applying Theorem 8 we obtain 119891119901(119909) equiv 1

119896(119909) = 119892

119896(119909)119901 119892

119901= 0 and 119891

119901(119909) = 119909

119902

Abstract and Applied Analysis 7

We will prove now by induction that deg(119891119895) le lfloor119895119902119901rfloor

and deg(119892119895) le lfloor119895119902119901rfloor minus 1 for all 119895 isin N and 119901 le 119895

For 119895 = 119901 is true since 119891119901(119909) = 119909

119902 and 119892119901

= 0 If wesuppose that the hypothesis is true for 1 le 119895 le 119895

0minus1 we have

to prove that deg(1198921198950) le lfloor119895

0119902119901rfloor minus 1 and deg(119891

1198950) le lfloor119895

0119902119901rfloor

On the other hand applying (28) we have

deg (1198921198950)

le max1le119895le1198950minus119901

deg (1198911198950minus119895

)+deg (119876119895+1

) deg (11989110158401198950minus119895

) + deg (119875119895)

le lfloor1198950119902

119901rfloor minus 1

(37)

since for 1 le 119895 le 1198950minus 119901 we have

deg (1198911198950minus119895

) + deg (119876119895+1

) le lfloor(1198950minus 119895) 119902119901rfloor + lfloor

119895119902

119901rfloor minus 1

le lfloor1198950119902

119901rfloor minus 1

deg (11989110158401198950minus119895

) + deg (119875119895) le lfloor

(1198950minus 119895) 119902

119901rfloor minus 1 + lfloor

119895119902

119901rfloor

le lfloor1198950119902

119901rfloor minus 1

(38)

By applying Lemma 10 we have that deg(1198911198950) le lfloor119895

0119902119901rfloorminus

1 In particular deg(1198911198950) le lfloor119895

0119902119901rfloor

Therefore if 119896 = 119872119901 deg(119892119896) le 119872119902 minus 1 119892119872119902

119896(0) = 0 and

119872119902

119896(0) = 0To finish the proof it is enough to apply Theorem 8

However as it is difficult to determine the new coordi-nates where system (19) has 119876

0(119909) equiv 119876

1(119909) equiv 0 we are

going to work with the original system (5) imposing directlythese two conditions In order that one of the series 119867

1or

1198672be a formal first integral of system (5) we have that these

series must have the term 119909119902119910119901 as a first monomial in its

development We will see that in this case and under someconditionswe can use the inductionmethod in order to verifythe existence of a formal first integral

The following result is only established for the series 1198671

but we can obtain a similar result for the case where 1198672is a

formal first integral of system (5)

Proposition 12 We consider system (5)where wewrite1198651and

1198652into the form

1198651(119909 119910) =

infin

sum

119894+119895=2

119886119894119895119909119894119910119895

1198652(119909 119910) =

infin

sum

119894+119895=2

119887119894119895119909119894119910119895

(39)

where 119886119894119895and 119887119894119895are arbitrary constants If we impose that 119867

1

be a first integral of system (5) we obtain that the first conditionis

infin

sum

119894=2

1198871198940

1199091198941198911(119909) + 119901119909119891

1015840

0(119909) +

infin

sum

119894=2

11988611989401199091198941198911015840

0(119909) = 0 (40)

The proof of this proposition is also straightforwarddeveloping in powers of 119910 and taking the coefficient of thepower 119910

0 Hence in order to have a recursive differentialequations where each 119891

119894(119909) does not depend on 119891

119894+1(119909) we

must impose that 1198871198940

= 0 for all 119894 and consequently 1198910must

be a constant that we can take without loss of generality equalto zero The consequence as before is that if all 119887

1198940= 0 for all

119894 the system (5) has 119910 as invariant algebraic curveThe next coefficients for each power of 119910 are of the form

119896(minus119902 +

infin

sum

119894=1

1198871198941119909119894)119891119896(119909)

+ (119901119909 +

infin

sum

119894=2

1198861198940119909119894)1198911015840

119896(119909) + 119892

119896(119909) = 0

(41)

where 119892119896(119909) depends of the previous119891

119894(119909) for 119894 = 1 2 119896minus

1 and for the first equation which corresponds to 119896 = 1 wehave that 119892

1(119909) = 0 Therefore 119891

1(119909) has the form

1198911(119909) = exp(minusint

minus119902 + suminfin

119894=11198871198941119909119894

119901119909 + suminfin

119894=21198861198940119909119894

) (42)

The integrand of the previous expression of 1198911is a rational

function which admits the following fraction decomposition

minus119902 + suminfin

119894=11198871198941

119909119894

119901119909 + suminfin

119894=21198861198940

119909119894=

119860

119909+

119861 (119909)

119901 + suminfin

119894=21198861198940

119909119894minus1 (43)

where 119860 = minus119902119901 and 119861(119909) is a formal series Therefore1198911(119909) = 119909

1199021199011198911(119909) where 119891

1is the corresponding formal

series obtained after the integration Now we consider theparticular case 119901 = 1 The following theorem gives sufficientconditions to have formal integrability in this case A similarproposition can be established for the case when the firstintegral is of the form119867

2

Proposition 13 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if 119901 = 1

1198871198940= 0 for 119894 ge 2 and 119887

1198941= 0 for 119894 ge 1 and

119892119896(119909) =

119896(119909) 119875 (119909 0)

1199091198911(119909)minus119896

with (119896119902)

119896(0) = 0 (44)

Proof Under the assumptions of the theorem we have that1198910= 0 119891

1= 1199091199021198911(119909) and (41) takes the form

minus119896119902119891119896(119909) + 119875 (119909 0) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (45)

whose solution is given by

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909 119904minus119896119902

1198911(119904)minus119896

119875 (119904 0)119892119896(119904) 119889119904)

(46)

8 Abstract and Applied Analysis

Taking into account the form of 119892119896(119909) we obtain

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909

119904minus1minus119896119902

119896(119904) 119889119904) (47)

Now it is straightforward to see that the integral does not givelogarithmic terms because the 119896119902-th derivative of

119896(119904) at zero

is zero that is (119896119902)119896

(0) = 0 We have that each 119891119896(119909) for all

119896 is an analytic function and therefore system (5) under theassumptions of the theorem admits a formal first integral ofthe form (39)

Examples where Proposition 13 can be applied can befound in [7 12 16 17] where some partial results are givenWe also can establish the following corollary in the casewherethe 119892119896(119909) are polynomials or some irrational functions

Corollary 14 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if one of

the following conditions holds

(a) 119901 = 1 1198861198940

= 1198871198940

= 0 for 119894 ge 2 and 1198871198941

= 0 for 119894 ge 1 and119892119896(119909) is a polynomial of degree at most 119896119902 minus 1

(b) 119901 = 1 1198861198940= 0 for 119894 = 119903 + 1 119887

1198940= 0 for 119894 ge 2 and 119887

1198941= 0

for 119894 ge 1with 119903 a positive integer and 119892119896(119909) is a rational

function of the form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119896119902119903

where 119896is a polynomial of degree at most 119896119902 minus 1

Proof In case (a) for we have1198910= 0 and119891

1= 119909119902 (41) take the

form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119892

119896(119909) = 0 (48)

whose solution is given by

119891119896(119909) = 119862

119896119909119896119902

minus 119909119896119902

int

119909

119904minus1minus119896119902

119892119896(119904) 119889119904 (49)

and taking into account that 119892119896(119909) that depends on 119891

119894for 119894 =

1 2 119896 minus 1 is at most of degree 119896119902 minus 1 we obtain that 119891119896is

a polynomial Moreover in this case system (5) becomes

= 119901119909 + 1199101206011(119909 119910)

= minus119902119910 + 1199101206012(119909 119910)

(50)

where 1206011and 120601

2are analytic functions

In case (b) we have that 1198910= 0 119891

1= 119909119902(1 + 119886119909

119903)minus119902119903 and

(41) takes the form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119886119909

119903+11198911015840

119896(119909) + 119892

119896(119909) = 0 (51)

whose solution is given by

119891119896(119909) = 119909

119896119902(1 + 119886119909

119903)minus119896119902119903

times (119862119896minus int

119909

119904minus1minus119896119902

(1 + 119886119904)minus1+119896119902119903

119892119896(119904) 119889119904)

(52)

and taking into account the form of 119892119896(119909) we obtain that

119891119896(119909) is also of the form 119891

119896(119909) = 119901

119896(119909)(1 + 119886119909

119903)minus119896119902119904 where

119901119896(119909) is a polynomial because the integral in (52) does not

give logarithmic terms

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separateform in order to be applied directly to different examplesCorollary 14 generalizes a lot of cases obtained in the liter-ature when some concrete families of polynomials systemswere studiedWe have notmade an exhaustive study of all thecases included in these large families but for instance containsthe cases (3) and (4) of Theorem 3 in [17] and case (3) ofTheorem 2 in [16]

In resume when we study a family that satisfies thenecessary conditions we must impose if the family systemhas a formal first integral of the form 119867

1or 1198672 If none of

them work we can look for the proper coordinates (1199111 1199112)

defined by the stable and instable manifold of the 119901 minus119902

resonant singular point In this way we must study if at leastone of the separatrices of the resonant point is algebraic Thebest case is when both are algebraicTherefore the problem tofind proper coordinates (119911

1 1199112) becomes a problem of finding

the invariant algebraic curves of system (5) passing throughthe 119901 minus119902 resonant singular point The existence of invariantalgebraic curves takes a leading role in the formal integrabilitytheory and is the base to find the proper coordinates systemThis is shown in the examples provided in [12] that aresolved by ad hoc methods and where the changes of variablesproposed to solve the examples are always invariant algebraiccurves of the original system passing through the originIn fact in [12] it recalled the Abhyankar-Moch theorem[37] This theorem establishes assuming the existence of aninvariant rational algebraic curve passing through the originthe existence of a rational invertible change of variables suchthat the invariant curve becomes one of the invariant axesIn this paper this result is generalized in the sense that any119901 minus119902 resonant singular point is formal orbitally equivalentto a system with 119910 = 0 as invariant algebraic curve seeLemma 5

6 Examples

In this section we present some examples where the resultsdeveloped in this paper are applied

Example 15 We first consider the 1 minus1 resonant quadraticsystem given by

= 119909 + 119886201199092+ 11988611119909119910 + 119886

021199102

= minus119910 + 119887201199092+ 11988711119909119910 + 119887

021199102

(53)

The aim is not to do an exhaustive study of when system (53)has a formal first integral in a neighborhood of the originIn fact the complex center cases were studied by Dulac see[34] The aim is only to show how to use the results givenin this paper If we apply directly Corollary 14 statement (a)we obtain 119887

20= 0 (this condition implies that 119910 = 0 be an

algebraic invariant curve of system (53)) and 11988620

= 11988711

= 0 Inthis case the quadratic system (53) reads

= 119909 + 11988611119909119910 + 119886

021199102

= minus119910 + 119887021199102

(54)

Abstract and Applied Analysis 9

and admits a formal first integral and by Theorem 2 system(54) has an analytic first integral in a neighborhood of theorigin

In fact the associated equation to system (54) is a linearequation with respect to 119909 and the system has a Darboux firstintegral of the form

119867(119909 119910)

= (11988702119910 minus 1)

minus1minus(1198861111988702)

times [211988602

minus 2 (1198860211988611

+ 1198860211988702) 119910

+ (1198863

11minus 119886111198872

02) 119909119910 + (119886

021198862

11+ 119886021198861111988702) 1199102]

(55)

Example 16 Consider the 1 minus1 resonant LotkandashVolterraplanar complex quartic system given by

= 119909 (1 minus 11988610119909 minus 11988601119910 minus 119886201199092minus 11988611119909119910

minus119886021199102minus 119886301199093minus 119886211199092119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988710119909 minus 11988701119910 minus 119887201199092minus 11988711119909119910

minus119887021199102minus 119887301199093minus 119887211199092119910 minus 119887121199091199102minus 119887031199103)

(56)

The LotkandashVolterra planar complex systems have been stud-ied in several papers see [7 11 13ndash15] In any case the LotkandashVolterra planar complex quartic system is an open problemcomputationally infeasible However with the results of thispaperwe are going to see that we can give some sufficient con-ditions to have an analytic first integral without computingany resonant focus value We propose a formal first integralof the form 119867

1 that is of the form (16) The recurrence that

gives119891119894(119909) is given by minus119894119891

119894(119909)+119909119891

1015840

119894(119909)+119892

119894(119909) = 0where 119892

119894(119909)

is(119894 minus 3) 119887

03119891119894minus3

(119909) + (119894 minus 2) 11988702119891119894minus2

(119909) + (119894 minus 2) 11988712119909119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) + (119894 minus 1) 11988711119909119891119894minus1

(119909)

+ (119894 minus 1) 119887211199092119891119894minus1

(119909)

+ 11989411988710119909119891119894(119909) + 119894119887

201199092119891119894(119909)

+ 119894119887301199093119891119894(119909) minus 119886

031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909) minus 119886

1111990921198911015840

119894minus1(119909)

minus 1198861011990921198911015840

119894(119909) minus 119886

2011990931198911015840

119894(119909) minus 119886

3011990941198911015840

119894(119909)

(57)

Now we impose that 119892119894(119909) must be of degree lei minus1 This

implies that 11988610

= 11988620

= 11988630

= 11988710

= 11988720

= 11988730

= 0 and11988721

= 11988621

= 11988711

= 11988611

= 0 and in this case the recurrence isminus119894119891119894(119909) + 119909119891

1015840

119894(119909) + 119892

119894(119909) = 0 where 119892

119894(119909) is given by

(119894 minus 3) 11988703119891119894minus3

(119909) + (119894 minus 2) (11988702

+ 11988712119909)119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) minus 119886031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909)

(58)

We obtain that its solution is given by

119891119894(119909) = 119862

119894119909119894minus 119909119894int

119909

119904minus1minus119894

119892119894(119904) 119889119904 (59)

and taking into account that 119892119894(119909) is of degree le 119894 minus 1 we can

establish the following result

Proposition 17 The system

= 119909 (1 minus 11988601119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988701119910 minus 119887021199102minus 119887121199091199102minus 119887031199103)

(60)

has an analytic first integral in a neighborhood of the origin

In fact Proposition 17 is a consequence of Proposition 11This technique can be applied to several families of polyno-mial systems and in fact gives a new mechanism to obtainresonant centers and also real centers for the complex systemsthat have a real preimage (when this preimage exists see eg[18])

Acknowledgments

A Algaba and C Garcıa are supported by aMICINNFEDERGrant no MTM2010-20907-C02-02 and by the Consejerıade Educacıon y Ciencia de la Junta de Andalucıa (projectsEXC2008FQM-872 TIC-130 and FQM-276) J Gine is par-tially supported by a MICINNFEDER Grant no MTM2011-22877 and by Generalitat de Catalunya Grant no 2009SGR381

References

[1] AAlgaba E Gamero andCGarcıa ldquoThe integrability problemfor a class of planar systemsrdquo Nonlinearity vol 22 no 2 pp395ndash420 2009

[2] A Algaba E Freire E Gamero and C Garcıa ldquoMonodromycenter-focus and integrability problems for quasi-homogeneouspolynomial systemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1726ndash1736 2010

[3] A Algaba N Fuentes and C Garcıa ldquoCenters of quasi-homogeneous polynomial planar systemsrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 419ndash431 2012

[4] A Algaba C Garcıa and M Reyes ldquoExistence of an inverseintegrating factor center problem and integrability of a class ofnilpotent systemsrdquo Chaos Solitons amp Fractals vol 45 no 6 pp869ndash878 2012

[5] J Chavarriga H Giacomini J Gine and J Llibre ldquoOn theintegrability of two-dimensional flowsrdquo Journal of DifferentialEquations vol 157 no 1 pp 163ndash182 1999

[6] H Zoładek ldquoThe problem of center for resonant singular pointsof polynomial vector fieldsrdquo Journal of Differential Equationsvol 137 no 1 pp 94ndash118 1997

[7] X Chen J Gine V G Romanovski and D S Shafer ldquoThe1 -q resonant center problem for certain cubic Lotka-Volterrasystemsrdquo Applied Mathematics and Computation vol 218 no23 pp 11620ndash11633 2012

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

4 The 120576-Method for Resonant Centers

We recall in this section the 120576-method developed in [23]which we apply here to resonant centers We consider system(2) which we write into the form

= 119875 (119909 119910) = 119901119909 + 1198651(119909 119910)

= 119876 (119909 119910) = minus119902 119910 + 1198652(119909 119910)

(5)

where 1198651(119909 119910) and 119865

2(119909 119910) are analytic functions without

constant and linear terms defined in a neighborhood of theorigin

To implement the algorithm we introduce a rescalingof the variables and a time rescaling given by (119909 119910 119905) rarr

(120576119901119909 120576119902119910 120576119903119905) where 120576 gt 0 and 119901 119902 and 119903 isin Z and system

(5) takes the form

= 120576119903minus119901

(119901 120576119901119909 + 1198651(120576119901119909 120576119902119910))

= 120576119903minus119902

(minus119902 120576119902119910 + 1198652(120576119901119909 120576119902119910))

(6)

We choose 119901 119902 119903 in such a way that system (6) will beanalytic in 120576 Hence by the classical theorem of the analyticdependence with respect to the parameters we have thatsystem (6) admits a first integral which can be developedin power of series of 120576 because it is analytic with respectto this parameter Therefore we can propose the followingdevelopment for the first integral

119867(119909 119910) =

infin

sum

119896=0

120576119896ℎ119896(119909 119910) (7)

where ℎ119896(119909 119910) are arbitrary functionsWenotice that119865

1(119909 119910)

and 1198652(119909 119910) are analytic functions where both can be null

and 1198651(0 0) = 119865

2(0 0) = 0 so we can develop them in a

neighborhood of the origin as convergent series of 119909 and 119910

of the form

1198651(119909 119910) = 119901

119899(119909 119910) + 119901

119899+1(119909 119910) + sdot sdot sdot + 119901

119895(119909 119910) + sdot sdot sdot

1198652(119909 119910) = 119902

119899(119909 119910) + 119902

119899+1(119909 119910) + sdot sdot sdot + 119902

119895(119909 119910) + sdot sdot sdot

(8)

with 119899 = minsubdeg|(00)

1198651(119909 119910) subdeg

|(00)1198652(119909 119910) ge 2

We recall that given an analytic function 119891(119909 119910)

defined in a neighborhood of a point (1199090 1199100) we define

subdeg|(1199090 1199100)

119891(119909 119910) as the least positive integer 119895 such thatsome derivative (120597119895119891120597119909119894120597119910119895minus119894)(119909

0 1199100) is not zero We notice

that this computation depends on the variables (119909 119910) onwhich the function 119891(119909 119910) depends so we will explicit thevariables used in each computation of subdeg For instancesubdeg

|(1199090 1199100)119891(119909 119910) = 0 if and only if 119891(119909

0 1199100) = 0 In (8)

119901119895(119909 119910) and 119902

119895(119909 119910) denote homogeneous polynomials of 119909

and 119910 of degree 119895 ge 119899 It is possible that 119901119899(119909 119910) or 119902

119899(119909 119910)

be null but by definition not both of them can be null Thesimplest case is to consider in the rescaling 119901 = 119902 = 1 Infact this case is equivalent to impose that system (8) has afirst integral which can be expanded as a formal series inhomogeneous parts

The richness of the 120576-method is that using the parameter120576 the functions ℎ

119896(119909 119910) need not be homogeneous parts

and we can construct also a singular series expansion in thevariables 119909 and 119910 see [23] The method depends heavily onℎ0the first integral of the initial quasi-homogeneous system

The simpler is ℎ0farther we go with the method However ℎ

0

can be chosen using different scalings of variables (119909 119910 119905) rarr

(120576119901119909 120576119902119910 120576119903119905) where 120576 gt 0 and 119901 119902 and 119903 isin Z in such

a way that ℎ0will be as simple as possible The method

gives necessary conditions to have analytic integrability ora singular series expansion around a singular point andinformation about what is called in [23] the essential variablesof a system In the method developed in [23] the parameter 120576needs not be small The parameter 120576 may be relatively large(for instance 120576 rarr 1) The convergence of series (7) withrespect to (119909 119910) must be analyzed in each particular caseand the convergent rate depends upon the nonlinear termsof the system (8) In the case of a resonant center the mostconvenient ℎ

0is 119909119902119910119901

From now on we suppose that system (5) admits aformal first integral Therefore ℎ

119896(119909 119910) are homogeneous

polynomials of degree 119896 and we can take ℎ0(119909 119910) = 0 In this

case if we impose that series (7) be a formal first integral ofsystem (5) we obtain at each order of homogeneity

119901119909120597ℎ119896

120597119909(119909 119910) minus 119902119910

120597ℎ119896

120597119910(119909 119910) + 119866

119896(119909 119910) = 0 (9)

where119866119896(119909 119910) depends on the previous ℎ

119894for 119894 = 1 119896minus1

The solution of the partial equation (9) is

ℎ119896(119909 119910) = 119865 (119909

119902119901119910) minus int

119909 119866119896(119904 (119909119902119901

119910) 119904119902119901

)

119901 119904119889119904 (10)

where 119865 is an arbitrary function of 119909119902119901119910 We require that ℎ119896

be a polynomial and therefore an ℎ119896not having logarithmic

terms In order to avoid the logarithmic terms it is easy to seethat the function119866

119896(119909 119910)must not have polynomial terms of

the form (119909119902119910119901)119898 for 119898 isin N This result is the same as that

we obtain using normal form theory see [6]We must impose the vanishing of all the logarithmic

terms (if there exists any difference from zero) and prove thatall the functions ℎ

119894(119909 119910) for 119894 gt 1 are polynomials In fact

we must prove this by induction assuming that all the ℎ119894for

119894 = 1 119896 minus 1 are polynomials and prove that the recursivepartial differential equation with respect to ℎ

119896(see (9)) gives

also a polynomial This is not so easy and even less workingwith a partial recursive differential equation as (9)

At this point we think in the blow-up 119911 = 119910119909 becauseeach ℎ

119894(119909 119910) are homogeneous polynomials of degree 119894

4 Abstract and Applied Analysis

This blowup transforms the system (5) into a system ofvariables (119909 119911) of the form

= minus (119901 + 119902) 119911 + 119909F (119909 119911)

= 119901119909 + 1199092G (119909 119911)

(11)

where F(0 0) = 0 After that we propose a formal firstintegral of the form

=

infin

sum

119894ge1

119891119894(119911) 119909119894 (12)

where 119891119894(119911) must be polynomials of degree lei (if the log-

arithmic terms are zero) If we impose that series (12) bea first integral of the system in variables (119909 119911) we obtainthat the term 119887

201199092 must be zero However we can vanish

this term doing the change 119910 = 119910 minus 1205721199092 and 119909 = 119909 with

120572 = minus11988720(2119901 + 119902) We must also take 119891

1(119911) = 0 Finally at

each power of 119910 we have the recursive equation

119896119901119891119896(119911) minus (119901 + 119902) 119911119891

1015840

119896(119911) + 119892

119896(119911) = 0 (13)

where 119892119896(119911) depends of the previous functions 119891

119894for 119894 =

1 119896 minus 1 Hence in this case instead of getting the partialrecursive differential equation (9) we obtain an ordinaryrecursive differential equation whose solution is given by

119891119896(119911) =

119896119911119896119901(119901+119902)

+ 119911119896119901(119901+119902)

int

119911 119904minus1minus119896119901(119901+119902)

119901 + 119902119892119896(119904) 119889119904

(14)

where 119896is an arbitrary constant However we are going to

see that although we place in a resonant center where all the119891119896be polynomials this blowup does not allow us to prove in

general by induction of the existence of a formal first integralof the form (12)

In this sense consider the recursive differential equation(13) whose solution is (14) and we assume that system (5) hasa resonant degenerate center at the origin It is easy to see thatthere always exists a value 119896

0such that for 119896 ge 119896

0the arbitrary

polynomials 119891119894(119911) for 119894 = 1 119896

0minus 1 can gives following the

recursive equation (13) logarithmic terms Therefore we cannot apply the induction method to prove that the recursiveequation (13) gives always a polynomial

To see this we must to see that the solution (14) of therecursive equation (13) can give a logarithmic term Thishappens when

minus1 minus1198960119901

119901 + 119902+ 119898119896= minus1 (15)

where119898119896is the degree of the polynomial 119892

119896(119904) From here we

have that 1198960= 119898119896(119901+119902)119901 Hence if119901 = 1 then 119896

0= 119898119896(1+119902)

which can be satisfied because119898119896and 119902 are positive integers

For the case 119901 = 1 and taking into account that the value of119898119896increases when 119896 increases it can also exist a value of119898

119896

such that 119898119896is divisible by 119901 and that gives the value of 119896

0

that can give logarithmic terms

Therefore the conclusion is that the formal constructionof the first integral (7) using homogeneous terms or using theblow-up 119911 = 119910119909 and the formal series (12) do not allow touse the induction method in order to verify the existence of aformal first integral We must use other developments whichis the subject of the next section

5 Other Developments of the FormalFirst Integral

In this section we consider other developments of the formalfirst integral We consider the formal development of the firstintegral of system (5) in a series in the variable 119909 or in 119910 thatis we consider

1198671=

infin

sum

119896=0

119891119896(119909) 119910119896 or 119867

2=

infin

sum

119896=0

119892119896(119910) 119909119896 (16)

First we consider a general analytic system that we canalways write into the following forms

= 119891 (119909) + 119910Φ1(119909 119910) = 119892 (119909) + 119910Φ

2(119909 119910) (17)

or

= 119891 (119910) + 119909Ψ1(119909 119910) = 119892 (119910) + 119909Ψ

2(119909 119910) (18)

where 119891 119892Φ1Φ2 Ψ1 and Ψ

2are analytic in their variables

For systems (17) and (18) we have the following straightfor-ward result

Proposition 4 If we impose that the series 1198671(1198672 resp) be

a first integral of system (17) (system (18) resp) among othersthe first condition is 1198911015840

0(119909)119891(119909) + 119891

1(119909)119892(119909) = 0 (1198911015840

0(119910)119892(119910) +

1198911(119910)119891(119910) = 0 resp)

In order that this condition generates a collection ofrecursive differential equations where each 119891

119894(119909) does not

depend on119891119894+1

(119909) wemust impose119892(119909) = 0 (119891(119910) = 0 resp)which implies that119910 = 0 (119909 = 0 resp) is an invariant algebraiccurve of system (17) (system (18) resp)

In fact for system (5) there are always a new coordinates(1199111 1199112) where 119911

1= 0 and 119911

2= 0 are invariant curves These

invariant curves are defined by the stable and instable man-ifold of the p minusq resonant singular point and therefore theprevious conditions are directly satisfied In these coordinatesany 119901 minus119902 resonant singular point is a Lotka-Volterra systemIn the following result we are going to see that we always canfind a new coordinate where 119910 = 0 is an invariant algebraiccurve of the transformed system (5)

We consider system (2) which we write into the form

= 119875 (119909 119910) = 119901119909 (1 + 1198750(119909)) + sum

119895ge1

119875119895(119909) 119910119895

= 119876 (119909 119910) = 1199091198760(119909) minus 119902119910 (1 + 119876

1(119909)) + sum

119895ge2

119876119895(119909) 119910119895

(19)

where 119875119895(119909) = sum

119894ge0119886119894119895119909119894 and 119876

119895(119909) = sum

119894ge0119887119894119895119909119894 are analytic

functions defined in a neighborhood of the origin

Abstract and Applied Analysis 5

Lemma 5 System (19) is orbitally equivalent to

= 119875 (119909 119910) = 119901119909 (1 + 1198750(119909)) + sum

119895ge1

119875119895(119909) 119910119895

= 119876 (119909 119910) = minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(20)

that is 1198760(119909) equiv 119876

1(119909) equiv 0 Moreover (119901 119902) are coprimes

Proof First we prove that we can achieve 1198760(119909) equiv 0 Assume

that 1198760(119909) equiv 0 and let 119873 = min119894 isin N cup 0 119887

1198940= 0 The

change of variable 119906 = 119909 V = 119910 minus (1198871198730

(119901(119873 + 1) + 119902))119909119873+1

transforms system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) V119895

V = 1199060(119906) minus 119902V (1 +

1(119906)) + sum

119895ge2

119895(119906) V119895

(21)

where 0(119906) = sum

119894ge01198871198940119906119894 and

1198940= 0 for 119894 = 0 1 119873

because V = minus ((119873 + 1)1198871198730

(119901(119873 + 1) + 119902))119906119873 hence

1199060(119906) = 119887

1198730119906119873+1

minus1199021198871198730

119901 (119873 + 1) + 119902119906119873+1

minus119901 (119873 + 1) 119887

1198730

119901 (119873 + 1) + 119902119906119873+1

+ O (119906119873+2

)

= O (119906119873+2

)

(22)

Therefore by means of successive change of variables we cantransform system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) 119910119895

V = minus119902V (1 + 1(119906)) + sum

119895ge2

119895(119906) V119895

(23)

To complete the proof it is enough to apply the scaling 119889119905 =

119889119879(1 + 1(119906))

Lemma 6 Let G be a formal function with G(0) = 0 Thesystem

(

) = (

119901119909

minus119902119910) + (

119901119909

119902119910)G (119909

119902119910119901) (24)

is formally integrable if and only ifG equiv 0

Proof The sufficient condition is trivial since 119909119902119910119901 is a first

integral of ( )119879 = (119901119909 minus119902119910)119879

We now will prove the necessary condition Let F(119909 119910) =(119901119909 minus119902119910)

119879+ (119901119909 119902119910)

119879G(ℎ) where ℎ = 119909

119902119910119901 and G(ℎ) =

sum119895ge1

119892119895ℎ119895 G equiv 0 otherwise the proof is finished We

consider 1198950= min119895 isin N | 119892

119895= 0 If 119868 is a first integral of

this system then there exists 119872 isin N such that 119868 = ℎ119872

+ sdot sdot sdot

and 0 = nabla119868 sdotF = 21199011199021198921198950ℎ1198950+1 + sdot sdot sdot Therefore 119892

1198950= 0 and this

is a contradiction

The next result shows that any integrable system (20)admits always a first integral of the form119867

1

Proposition 7 If system (20) is formally integrable then119867(119909 119910) = sum

119895ge0119891119895(119909)119910119895 with 119891

119895(119909) a formal function is a first

integral and moreover 1198910= sdot sdot sdot 119891

119901minus1= 0

Proof By applying the Poincare-Dulac normal form thereexists a change of variable to transform (20) into

(

V) = (

119901119906

minus119902V) + (

sum

119896ge1

119886119896119906ℎ119896

sum

119896ge1

119887119896Vℎ119896) (25)

where ℎ = 119906119902V119901 We can assume that the change of variable is

119906 = 119909 + sdot sdot sdot V = 119910(1 + sdot sdot sdot) because the axis 119910 = 0 and V = 0

are invariantConsidering the formal functions F(ℎ) = sum

119896ge1119888119896ℎ119896 and

G(ℎ) = sum119896ge1

119889119896ℎ119896 with 119888

119896= (119902119886119896minus 119901119887119896)2119901119902 and 119889

119896= (119902119886119896+

119901119887119896)2119901119902 we get

(

V) = (

119901119906

minus119902V) (1 +F (119906

119902V119901)) + (

119901119906

119902V) G (119906

119902V119901) (26)

whereF G are formal functions withF(0) = G(0) = 0Moreover by the scaling of the time 119889119905 = (1 +F)119889120591 we

can getF equiv 0 that is it is possible to transform (20) into

(1199061015840

V1015840) = (

119901119906

minus119902V) + (

119901119906

119902V)G (119906

119902V119901) (27)

whereG = G(1 +F) is a formal function withG(0) = 0If this system is formally integrable applying Lemma 6

we obtainG = 0Therefore119867(119909 119910) = (119909+sdot sdot sdot )119902119910119901(1+sdot sdot sdot )

119901=

sum119896ge119901

119891119896(119909)119910119896 is a first integral of system (20)

The main result of this work is as follows

Theorem 8 System (20) admits an analytic first integral119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896 if for each 119896 isin N such that 119896 = 119872119901

with119872 isin N is verified that the derivative (119872119902)119896

(0) = 0 where119896for 119896 ge 119901 is the analytical function

119896(119909) =

119892119896(119909)

119901 (1 + 1198750(119909)) 119891

119901(119909)119896

119891119901(119909) = exp(minusint

119909

0

(1199021198750(119904)

119901119904 (1 + 1198750(119904))

) 119889119904)

119892119901= 0 and for 119896 gt 119901

119892119896(119909) =

119896minus119901

sum

119895=1

((119896 minus 119895) 119891119896minus119895

(119909)119876119895+1

(119909) + 1198911015840

119896minus119895(119909) 119875119895(119909))

(28)

and for 119896 ge 119901

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

119904minus1minus(119896119902119901)

119896(119904) 119889119904) (29)

6 Abstract and Applied Analysis

Proof By Proposition 7 we have 1198910

= sdot sdot sdot = 119891119901minus1

= 0 Ifwe impose that 119867(119909 119910) be a first integral of system (20) weobtain that the first condition is

minus119901119902119891119901(119909) + 119901119909119891

1015840

119901(119909) (1 + 119875

0(119909)) = 0 (30)

The next coefficients for each power of 119910 are of the form

minus119902119896119891119896(119909) + 119901119909 (1 + 119875

0(119909)) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (31)

where 119892119901= 0 and 119892

119896(119909) for 119896 gt 119901 depends of the previous

119891119894(119909) for 119894 = 119901 119901 + 1 119896 minus 1 More specifically we obtain

the expression (28)The solution of homogeneous equation associated to (31)

is

119891ℎ

119896(119909) = exp(int

119896119902

119901119909 (1 + 1198750(119909))

) (32)

We can define the following analytical function 119887(119909) =

minus(119902119901119909)(1198750(119909)(1+119875

0(119909))) and the integrand of the previous

expression of 119891ℎ

119896is a rational function which admits the

following fraction decomposition

119896119902

119901119909 (1 + 1198750(119909))

=119896119902

119901119909+ 119896119887 (119909) (33)

Therefore 119891ℎ

119896(119909) = 119909

119896119902119901 exp(int 119896119887(119909)) =

119909119896119902119901

(exp(int1199090119887(119904)119889119904))

119896

and for the first equation whichcorresponds to 119896 = 119901 we have that 119892

119901(119909) = 0 Therefore

119891ℎ

119901(119909) has the form 119891

ℎ

119901(119909) = 119862

119901119909119902(119891119901(119909))119901

where 119891119901(119909) =

exp(int1199090119887(119904)119889119904) is an analytic function with 119891

119901(0) = 1 So

119891ℎ

119896(119909) = 119862

119896119909119896119902119901

119891119901(119909)119896 and the solution of (31) is given by

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

0

119904minus119896119902119901

119891119901(119904)minus119896

119901119904 (1 + 1198750(119904))

119892119896(119904) 119889119904)

(34)

Taking into account the form of 119892119896(119909) we obtain the

expression (29) for119891119896 Now it is straightforward to see that the

integral does not give logarithmic terms in the case 119896119902119901 notin N

and in the case 119896 = 119872119901 because (119872119902)

119896(0) = 0 If we choose

119862119896

= 0 we have that each 119891119896(119909) for all 119896 is an analytic

function and therefore system (20) under the assumptionsof the theorem admits a formal first integral of the form119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896

Corollary 9 System (20) admits a formal first integral of theform119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895 if one of the following conditions

holds

(a) 1198750(119909) equiv 0 and for each 119896 = 119872119901 119872 isin N 119892

119896(119909) is a

polynomial of degree at most119872119902 minus 1(b) 1198750(119909) equiv 119886119909

119903 with 119903 a positive integer and for each119896 = 119872119901 119872 isin N 119892

119896(119909) is a rational function of the

form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119902119903119901 where

119896is a

polynomial of degree at most119872119902 minus 1

Proof In case (a) it is enough to applyTheorem 8 for119891119901(119909) equiv

1 and 119896(119909) = 119892

119896(119909)119901 For 119896 = 119872119901 119872 isin N and we have

(119872119902)

119896(119909) = 119892

(119872119902)

119896119901 = 0 because the degree of 119892

119896(119909) is less

than 119902119872In case (b) we have that 119891

119901(119909) equiv (1 + 119886119909

119903)minus119902119903119901 and

119896(119909) = 119892

119896(119909)119901(1 + 119886119909

119903)1minus119902119903119901 Therefore

119896(119909) =

119896(119909)119901

and (119872119901)

119896(119909) = 0 because the degree of

119896(119909) is less than 119902119872

We obtain the result by applying also Theorem 8

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separate formin order to be applied directly to different examples

Lemma 10 If system (20)with 1198750(119909) equiv 0 admits a formal first

integral of the form119867(119909 119910) = sum119895ge119901

119891119895(119909)119910119895 then the following

conditions hold

(a) If for each 119896 such that 119896119902119901 notin N 119892119896is a polyno-

mial then it is possible to choose 119891119896polynomial with

deg(119891119896) = deg(119892

119896)

(b) If for each 119896 = 119872119901 119872 isin N 119892119896(119909) is a polynomial

with deg(119892119896) le 119872119902 minus 1 then it is possible to choose 119891

119896

polynomial with deg(119891119896) le 119872119902 minus 1

Proof By applying Theorem 8 we obtain 119891119901(119909) equiv 1 and

119896(119909) = 119892

119896(119909)119901 If we choose 119862

119896= 0 in (29) we obtain

119891119896(119909) = minus

1

119901119909119896119902119901

int

119909

119904minus1minus119896119902119901

119892119896(119904) 119889119904 (35)

In case (a) the result is trivial since the integral does not givelogarithmic terms because 119896119902119901 notin N

In case (b) 119896 = 119872119901 and 119891119896(119909) =

minus(1119901)119909119872119902

int119909

119904minus1minus119872119902

119892119896(119904) 119889119904Therefore we obtain the result

applying that 119892119896is a polynomial with deg(119892

119896) le 119872119902 minus 1

Based on the results presented in this work the followingproposition gives a large family of analytic systems that havean analytic first integral

Proposition 11 The analytic system

= 119901119909 + sum

119895ge1

119875119895(119909) 119910119895

= minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(36)

where 119875119895and 119876

119895are polynomials such that deg(119875

119895) le lfloor119895119902119901rfloor

deg(119876119895) le lfloor(119895 minus 1)119902119901rfloor minus 1 has an analytic first integral in

a neighborhood of the origin where lfloor119886rfloor is the integer part of119886 isin R

Proof From Proposition 7 if system (36) is integrable thena first integral is of the form 119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895

where 119891119895(119909) is a formal function Moreover 119875

0(119909) equiv 0 and

therefore by applying Theorem 8 we obtain 119891119901(119909) equiv 1

119896(119909) = 119892

119896(119909)119901 119892

119901= 0 and 119891

119901(119909) = 119909

119902

Abstract and Applied Analysis 7

We will prove now by induction that deg(119891119895) le lfloor119895119902119901rfloor

and deg(119892119895) le lfloor119895119902119901rfloor minus 1 for all 119895 isin N and 119901 le 119895

For 119895 = 119901 is true since 119891119901(119909) = 119909

119902 and 119892119901

= 0 If wesuppose that the hypothesis is true for 1 le 119895 le 119895

0minus1 we have

to prove that deg(1198921198950) le lfloor119895

0119902119901rfloor minus 1 and deg(119891

1198950) le lfloor119895

0119902119901rfloor

On the other hand applying (28) we have

deg (1198921198950)

le max1le119895le1198950minus119901

deg (1198911198950minus119895

)+deg (119876119895+1

) deg (11989110158401198950minus119895

) + deg (119875119895)

le lfloor1198950119902

119901rfloor minus 1

(37)

since for 1 le 119895 le 1198950minus 119901 we have

deg (1198911198950minus119895

) + deg (119876119895+1

) le lfloor(1198950minus 119895) 119902119901rfloor + lfloor

119895119902

119901rfloor minus 1

le lfloor1198950119902

119901rfloor minus 1

deg (11989110158401198950minus119895

) + deg (119875119895) le lfloor

(1198950minus 119895) 119902

119901rfloor minus 1 + lfloor

119895119902

119901rfloor

le lfloor1198950119902

119901rfloor minus 1

(38)

By applying Lemma 10 we have that deg(1198911198950) le lfloor119895

0119902119901rfloorminus

1 In particular deg(1198911198950) le lfloor119895

0119902119901rfloor

Therefore if 119896 = 119872119901 deg(119892119896) le 119872119902 minus 1 119892119872119902

119896(0) = 0 and

119872119902

119896(0) = 0To finish the proof it is enough to apply Theorem 8

However as it is difficult to determine the new coordi-nates where system (19) has 119876

0(119909) equiv 119876

1(119909) equiv 0 we are

going to work with the original system (5) imposing directlythese two conditions In order that one of the series 119867

1or

1198672be a formal first integral of system (5) we have that these

series must have the term 119909119902119910119901 as a first monomial in its

development We will see that in this case and under someconditionswe can use the inductionmethod in order to verifythe existence of a formal first integral

The following result is only established for the series 1198671

but we can obtain a similar result for the case where 1198672is a

formal first integral of system (5)

Proposition 12 We consider system (5)where wewrite1198651and

1198652into the form

1198651(119909 119910) =

infin

sum

119894+119895=2

119886119894119895119909119894119910119895

1198652(119909 119910) =

infin

sum

119894+119895=2

119887119894119895119909119894119910119895

(39)

where 119886119894119895and 119887119894119895are arbitrary constants If we impose that 119867

1

be a first integral of system (5) we obtain that the first conditionis

infin

sum

119894=2

1198871198940

1199091198941198911(119909) + 119901119909119891

1015840

0(119909) +

infin

sum

119894=2

11988611989401199091198941198911015840

0(119909) = 0 (40)

The proof of this proposition is also straightforwarddeveloping in powers of 119910 and taking the coefficient of thepower 119910

0 Hence in order to have a recursive differentialequations where each 119891

119894(119909) does not depend on 119891

119894+1(119909) we

must impose that 1198871198940

= 0 for all 119894 and consequently 1198910must

be a constant that we can take without loss of generality equalto zero The consequence as before is that if all 119887

1198940= 0 for all

119894 the system (5) has 119910 as invariant algebraic curveThe next coefficients for each power of 119910 are of the form

119896(minus119902 +

infin

sum

119894=1

1198871198941119909119894)119891119896(119909)

+ (119901119909 +

infin

sum

119894=2

1198861198940119909119894)1198911015840

119896(119909) + 119892

119896(119909) = 0

(41)

where 119892119896(119909) depends of the previous119891

119894(119909) for 119894 = 1 2 119896minus

1 and for the first equation which corresponds to 119896 = 1 wehave that 119892

1(119909) = 0 Therefore 119891

1(119909) has the form

1198911(119909) = exp(minusint

minus119902 + suminfin

119894=11198871198941119909119894

119901119909 + suminfin

119894=21198861198940119909119894

) (42)

The integrand of the previous expression of 1198911is a rational

function which admits the following fraction decomposition

minus119902 + suminfin

119894=11198871198941

119909119894

119901119909 + suminfin

119894=21198861198940

119909119894=

119860

119909+

119861 (119909)

119901 + suminfin

119894=21198861198940

119909119894minus1 (43)

where 119860 = minus119902119901 and 119861(119909) is a formal series Therefore1198911(119909) = 119909

1199021199011198911(119909) where 119891

1is the corresponding formal

series obtained after the integration Now we consider theparticular case 119901 = 1 The following theorem gives sufficientconditions to have formal integrability in this case A similarproposition can be established for the case when the firstintegral is of the form119867

2

Proposition 13 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if 119901 = 1

1198871198940= 0 for 119894 ge 2 and 119887

1198941= 0 for 119894 ge 1 and

119892119896(119909) =

119896(119909) 119875 (119909 0)

1199091198911(119909)minus119896

with (119896119902)

119896(0) = 0 (44)

Proof Under the assumptions of the theorem we have that1198910= 0 119891

1= 1199091199021198911(119909) and (41) takes the form

minus119896119902119891119896(119909) + 119875 (119909 0) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (45)

whose solution is given by

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909 119904minus119896119902

1198911(119904)minus119896

119875 (119904 0)119892119896(119904) 119889119904)

(46)

8 Abstract and Applied Analysis

Taking into account the form of 119892119896(119909) we obtain

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909

119904minus1minus119896119902

119896(119904) 119889119904) (47)

Now it is straightforward to see that the integral does not givelogarithmic terms because the 119896119902-th derivative of

119896(119904) at zero

is zero that is (119896119902)119896

(0) = 0 We have that each 119891119896(119909) for all

119896 is an analytic function and therefore system (5) under theassumptions of the theorem admits a formal first integral ofthe form (39)

Examples where Proposition 13 can be applied can befound in [7 12 16 17] where some partial results are givenWe also can establish the following corollary in the casewherethe 119892119896(119909) are polynomials or some irrational functions

Corollary 14 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if one of

the following conditions holds

(a) 119901 = 1 1198861198940

= 1198871198940

= 0 for 119894 ge 2 and 1198871198941

= 0 for 119894 ge 1 and119892119896(119909) is a polynomial of degree at most 119896119902 minus 1

(b) 119901 = 1 1198861198940= 0 for 119894 = 119903 + 1 119887

1198940= 0 for 119894 ge 2 and 119887

1198941= 0

for 119894 ge 1with 119903 a positive integer and 119892119896(119909) is a rational

function of the form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119896119902119903

where 119896is a polynomial of degree at most 119896119902 minus 1

Proof In case (a) for we have1198910= 0 and119891

1= 119909119902 (41) take the

form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119892

119896(119909) = 0 (48)

whose solution is given by

119891119896(119909) = 119862

119896119909119896119902

minus 119909119896119902

int

119909

119904minus1minus119896119902

119892119896(119904) 119889119904 (49)

and taking into account that 119892119896(119909) that depends on 119891

119894for 119894 =

1 2 119896 minus 1 is at most of degree 119896119902 minus 1 we obtain that 119891119896is

a polynomial Moreover in this case system (5) becomes

= 119901119909 + 1199101206011(119909 119910)

= minus119902119910 + 1199101206012(119909 119910)

(50)

where 1206011and 120601

2are analytic functions

In case (b) we have that 1198910= 0 119891

1= 119909119902(1 + 119886119909

119903)minus119902119903 and

(41) takes the form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119886119909

119903+11198911015840

119896(119909) + 119892

119896(119909) = 0 (51)

whose solution is given by

119891119896(119909) = 119909

119896119902(1 + 119886119909

119903)minus119896119902119903

times (119862119896minus int

119909

119904minus1minus119896119902

(1 + 119886119904)minus1+119896119902119903

119892119896(119904) 119889119904)

(52)

and taking into account the form of 119892119896(119909) we obtain that

119891119896(119909) is also of the form 119891

119896(119909) = 119901

119896(119909)(1 + 119886119909

119903)minus119896119902119904 where

119901119896(119909) is a polynomial because the integral in (52) does not

give logarithmic terms

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separateform in order to be applied directly to different examplesCorollary 14 generalizes a lot of cases obtained in the liter-ature when some concrete families of polynomials systemswere studiedWe have notmade an exhaustive study of all thecases included in these large families but for instance containsthe cases (3) and (4) of Theorem 3 in [17] and case (3) ofTheorem 2 in [16]

In resume when we study a family that satisfies thenecessary conditions we must impose if the family systemhas a formal first integral of the form 119867

1or 1198672 If none of

them work we can look for the proper coordinates (1199111 1199112)

defined by the stable and instable manifold of the 119901 minus119902

resonant singular point In this way we must study if at leastone of the separatrices of the resonant point is algebraic Thebest case is when both are algebraicTherefore the problem tofind proper coordinates (119911

1 1199112) becomes a problem of finding

the invariant algebraic curves of system (5) passing throughthe 119901 minus119902 resonant singular point The existence of invariantalgebraic curves takes a leading role in the formal integrabilitytheory and is the base to find the proper coordinates systemThis is shown in the examples provided in [12] that aresolved by ad hoc methods and where the changes of variablesproposed to solve the examples are always invariant algebraiccurves of the original system passing through the originIn fact in [12] it recalled the Abhyankar-Moch theorem[37] This theorem establishes assuming the existence of aninvariant rational algebraic curve passing through the originthe existence of a rational invertible change of variables suchthat the invariant curve becomes one of the invariant axesIn this paper this result is generalized in the sense that any119901 minus119902 resonant singular point is formal orbitally equivalentto a system with 119910 = 0 as invariant algebraic curve seeLemma 5

6 Examples

In this section we present some examples where the resultsdeveloped in this paper are applied

Example 15 We first consider the 1 minus1 resonant quadraticsystem given by

= 119909 + 119886201199092+ 11988611119909119910 + 119886

021199102

= minus119910 + 119887201199092+ 11988711119909119910 + 119887

021199102

(53)

The aim is not to do an exhaustive study of when system (53)has a formal first integral in a neighborhood of the originIn fact the complex center cases were studied by Dulac see[34] The aim is only to show how to use the results givenin this paper If we apply directly Corollary 14 statement (a)we obtain 119887

20= 0 (this condition implies that 119910 = 0 be an

algebraic invariant curve of system (53)) and 11988620

= 11988711

= 0 Inthis case the quadratic system (53) reads

= 119909 + 11988611119909119910 + 119886

021199102

= minus119910 + 119887021199102

(54)

Abstract and Applied Analysis 9

and admits a formal first integral and by Theorem 2 system(54) has an analytic first integral in a neighborhood of theorigin

In fact the associated equation to system (54) is a linearequation with respect to 119909 and the system has a Darboux firstintegral of the form

119867(119909 119910)

= (11988702119910 minus 1)

minus1minus(1198861111988702)

times [211988602

minus 2 (1198860211988611

+ 1198860211988702) 119910

+ (1198863

11minus 119886111198872

02) 119909119910 + (119886

021198862

11+ 119886021198861111988702) 1199102]

(55)

Example 16 Consider the 1 minus1 resonant LotkandashVolterraplanar complex quartic system given by

= 119909 (1 minus 11988610119909 minus 11988601119910 minus 119886201199092minus 11988611119909119910

minus119886021199102minus 119886301199093minus 119886211199092119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988710119909 minus 11988701119910 minus 119887201199092minus 11988711119909119910

minus119887021199102minus 119887301199093minus 119887211199092119910 minus 119887121199091199102minus 119887031199103)

(56)

The LotkandashVolterra planar complex systems have been stud-ied in several papers see [7 11 13ndash15] In any case the LotkandashVolterra planar complex quartic system is an open problemcomputationally infeasible However with the results of thispaperwe are going to see that we can give some sufficient con-ditions to have an analytic first integral without computingany resonant focus value We propose a formal first integralof the form 119867

1 that is of the form (16) The recurrence that

gives119891119894(119909) is given by minus119894119891

119894(119909)+119909119891

1015840

119894(119909)+119892

119894(119909) = 0where 119892

119894(119909)

is(119894 minus 3) 119887

03119891119894minus3

(119909) + (119894 minus 2) 11988702119891119894minus2

(119909) + (119894 minus 2) 11988712119909119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) + (119894 minus 1) 11988711119909119891119894minus1

(119909)

+ (119894 minus 1) 119887211199092119891119894minus1

(119909)

+ 11989411988710119909119891119894(119909) + 119894119887

201199092119891119894(119909)

+ 119894119887301199093119891119894(119909) minus 119886

031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909) minus 119886

1111990921198911015840

119894minus1(119909)

minus 1198861011990921198911015840

119894(119909) minus 119886

2011990931198911015840

119894(119909) minus 119886

3011990941198911015840

119894(119909)

(57)

Now we impose that 119892119894(119909) must be of degree lei minus1 This

implies that 11988610

= 11988620

= 11988630

= 11988710

= 11988720

= 11988730

= 0 and11988721

= 11988621

= 11988711

= 11988611

= 0 and in this case the recurrence isminus119894119891119894(119909) + 119909119891

1015840

119894(119909) + 119892

119894(119909) = 0 where 119892

119894(119909) is given by

(119894 minus 3) 11988703119891119894minus3

(119909) + (119894 minus 2) (11988702

+ 11988712119909)119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) minus 119886031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909)

(58)

We obtain that its solution is given by

119891119894(119909) = 119862

119894119909119894minus 119909119894int

119909

119904minus1minus119894

119892119894(119904) 119889119904 (59)

and taking into account that 119892119894(119909) is of degree le 119894 minus 1 we can

establish the following result

Proposition 17 The system

= 119909 (1 minus 11988601119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988701119910 minus 119887021199102minus 119887121199091199102minus 119887031199103)

(60)

has an analytic first integral in a neighborhood of the origin

In fact Proposition 17 is a consequence of Proposition 11This technique can be applied to several families of polyno-mial systems and in fact gives a new mechanism to obtainresonant centers and also real centers for the complex systemsthat have a real preimage (when this preimage exists see eg[18])

Acknowledgments

A Algaba and C Garcıa are supported by aMICINNFEDERGrant no MTM2010-20907-C02-02 and by the Consejerıade Educacıon y Ciencia de la Junta de Andalucıa (projectsEXC2008FQM-872 TIC-130 and FQM-276) J Gine is par-tially supported by a MICINNFEDER Grant no MTM2011-22877 and by Generalitat de Catalunya Grant no 2009SGR381

References

[1] AAlgaba E Gamero andCGarcıa ldquoThe integrability problemfor a class of planar systemsrdquo Nonlinearity vol 22 no 2 pp395ndash420 2009

[2] A Algaba E Freire E Gamero and C Garcıa ldquoMonodromycenter-focus and integrability problems for quasi-homogeneouspolynomial systemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1726ndash1736 2010

[3] A Algaba N Fuentes and C Garcıa ldquoCenters of quasi-homogeneous polynomial planar systemsrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 419ndash431 2012

[4] A Algaba C Garcıa and M Reyes ldquoExistence of an inverseintegrating factor center problem and integrability of a class ofnilpotent systemsrdquo Chaos Solitons amp Fractals vol 45 no 6 pp869ndash878 2012

[5] J Chavarriga H Giacomini J Gine and J Llibre ldquoOn theintegrability of two-dimensional flowsrdquo Journal of DifferentialEquations vol 157 no 1 pp 163ndash182 1999

[6] H Zoładek ldquoThe problem of center for resonant singular pointsof polynomial vector fieldsrdquo Journal of Differential Equationsvol 137 no 1 pp 94ndash118 1997

[7] X Chen J Gine V G Romanovski and D S Shafer ldquoThe1 -q resonant center problem for certain cubic Lotka-Volterrasystemsrdquo Applied Mathematics and Computation vol 218 no23 pp 11620ndash11633 2012

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

This blowup transforms the system (5) into a system ofvariables (119909 119911) of the form

= minus (119901 + 119902) 119911 + 119909F (119909 119911)

= 119901119909 + 1199092G (119909 119911)

(11)

where F(0 0) = 0 After that we propose a formal firstintegral of the form

=

infin

sum

119894ge1

119891119894(119911) 119909119894 (12)

where 119891119894(119911) must be polynomials of degree lei (if the log-

arithmic terms are zero) If we impose that series (12) bea first integral of the system in variables (119909 119911) we obtainthat the term 119887

201199092 must be zero However we can vanish

this term doing the change 119910 = 119910 minus 1205721199092 and 119909 = 119909 with

120572 = minus11988720(2119901 + 119902) We must also take 119891

1(119911) = 0 Finally at

each power of 119910 we have the recursive equation

119896119901119891119896(119911) minus (119901 + 119902) 119911119891

1015840

119896(119911) + 119892

119896(119911) = 0 (13)

where 119892119896(119911) depends of the previous functions 119891

119894for 119894 =

1 119896 minus 1 Hence in this case instead of getting the partialrecursive differential equation (9) we obtain an ordinaryrecursive differential equation whose solution is given by

119891119896(119911) =

119896119911119896119901(119901+119902)

+ 119911119896119901(119901+119902)

int

119911 119904minus1minus119896119901(119901+119902)

119901 + 119902119892119896(119904) 119889119904

(14)

where 119896is an arbitrary constant However we are going to

see that although we place in a resonant center where all the119891119896be polynomials this blowup does not allow us to prove in

general by induction of the existence of a formal first integralof the form (12)

In this sense consider the recursive differential equation(13) whose solution is (14) and we assume that system (5) hasa resonant degenerate center at the origin It is easy to see thatthere always exists a value 119896

0such that for 119896 ge 119896

0the arbitrary

polynomials 119891119894(119911) for 119894 = 1 119896

0minus 1 can gives following the

recursive equation (13) logarithmic terms Therefore we cannot apply the induction method to prove that the recursiveequation (13) gives always a polynomial

To see this we must to see that the solution (14) of therecursive equation (13) can give a logarithmic term Thishappens when

minus1 minus1198960119901

119901 + 119902+ 119898119896= minus1 (15)

where119898119896is the degree of the polynomial 119892

119896(119904) From here we

have that 1198960= 119898119896(119901+119902)119901 Hence if119901 = 1 then 119896

0= 119898119896(1+119902)

which can be satisfied because119898119896and 119902 are positive integers

For the case 119901 = 1 and taking into account that the value of119898119896increases when 119896 increases it can also exist a value of119898

119896

such that 119898119896is divisible by 119901 and that gives the value of 119896

0

that can give logarithmic terms

Therefore the conclusion is that the formal constructionof the first integral (7) using homogeneous terms or using theblow-up 119911 = 119910119909 and the formal series (12) do not allow touse the induction method in order to verify the existence of aformal first integral We must use other developments whichis the subject of the next section

5 Other Developments of the FormalFirst Integral

In this section we consider other developments of the formalfirst integral We consider the formal development of the firstintegral of system (5) in a series in the variable 119909 or in 119910 thatis we consider

1198671=

infin

sum

119896=0

119891119896(119909) 119910119896 or 119867

2=

infin

sum

119896=0

119892119896(119910) 119909119896 (16)

First we consider a general analytic system that we canalways write into the following forms

= 119891 (119909) + 119910Φ1(119909 119910) = 119892 (119909) + 119910Φ

2(119909 119910) (17)

or

= 119891 (119910) + 119909Ψ1(119909 119910) = 119892 (119910) + 119909Ψ

2(119909 119910) (18)

where 119891 119892Φ1Φ2 Ψ1 and Ψ

2are analytic in their variables

For systems (17) and (18) we have the following straightfor-ward result

Proposition 4 If we impose that the series 1198671(1198672 resp) be

a first integral of system (17) (system (18) resp) among othersthe first condition is 1198911015840

0(119909)119891(119909) + 119891

1(119909)119892(119909) = 0 (1198911015840

0(119910)119892(119910) +

1198911(119910)119891(119910) = 0 resp)

In order that this condition generates a collection ofrecursive differential equations where each 119891

119894(119909) does not

depend on119891119894+1

(119909) wemust impose119892(119909) = 0 (119891(119910) = 0 resp)which implies that119910 = 0 (119909 = 0 resp) is an invariant algebraiccurve of system (17) (system (18) resp)

In fact for system (5) there are always a new coordinates(1199111 1199112) where 119911

1= 0 and 119911

2= 0 are invariant curves These

invariant curves are defined by the stable and instable man-ifold of the p minusq resonant singular point and therefore theprevious conditions are directly satisfied In these coordinatesany 119901 minus119902 resonant singular point is a Lotka-Volterra systemIn the following result we are going to see that we always canfind a new coordinate where 119910 = 0 is an invariant algebraiccurve of the transformed system (5)

We consider system (2) which we write into the form

= 119875 (119909 119910) = 119901119909 (1 + 1198750(119909)) + sum

119895ge1

119875119895(119909) 119910119895

= 119876 (119909 119910) = 1199091198760(119909) minus 119902119910 (1 + 119876

1(119909)) + sum

119895ge2

119876119895(119909) 119910119895

(19)

where 119875119895(119909) = sum

119894ge0119886119894119895119909119894 and 119876

119895(119909) = sum

119894ge0119887119894119895119909119894 are analytic

functions defined in a neighborhood of the origin

Abstract and Applied Analysis 5

Lemma 5 System (19) is orbitally equivalent to

= 119875 (119909 119910) = 119901119909 (1 + 1198750(119909)) + sum

119895ge1

119875119895(119909) 119910119895

= 119876 (119909 119910) = minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(20)

that is 1198760(119909) equiv 119876

1(119909) equiv 0 Moreover (119901 119902) are coprimes

Proof First we prove that we can achieve 1198760(119909) equiv 0 Assume

that 1198760(119909) equiv 0 and let 119873 = min119894 isin N cup 0 119887

1198940= 0 The

change of variable 119906 = 119909 V = 119910 minus (1198871198730

(119901(119873 + 1) + 119902))119909119873+1

transforms system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) V119895

V = 1199060(119906) minus 119902V (1 +

1(119906)) + sum

119895ge2

119895(119906) V119895

(21)

where 0(119906) = sum

119894ge01198871198940119906119894 and

1198940= 0 for 119894 = 0 1 119873

because V = minus ((119873 + 1)1198871198730

(119901(119873 + 1) + 119902))119906119873 hence

1199060(119906) = 119887

1198730119906119873+1

minus1199021198871198730

119901 (119873 + 1) + 119902119906119873+1

minus119901 (119873 + 1) 119887

1198730

119901 (119873 + 1) + 119902119906119873+1

+ O (119906119873+2

)

= O (119906119873+2

)

(22)

Therefore by means of successive change of variables we cantransform system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) 119910119895

V = minus119902V (1 + 1(119906)) + sum

119895ge2

119895(119906) V119895

(23)

To complete the proof it is enough to apply the scaling 119889119905 =

119889119879(1 + 1(119906))

Lemma 6 Let G be a formal function with G(0) = 0 Thesystem

(

) = (

119901119909

minus119902119910) + (

119901119909

119902119910)G (119909

119902119910119901) (24)

is formally integrable if and only ifG equiv 0

Proof The sufficient condition is trivial since 119909119902119910119901 is a first

integral of ( )119879 = (119901119909 minus119902119910)119879

We now will prove the necessary condition Let F(119909 119910) =(119901119909 minus119902119910)

119879+ (119901119909 119902119910)

119879G(ℎ) where ℎ = 119909

119902119910119901 and G(ℎ) =

sum119895ge1

119892119895ℎ119895 G equiv 0 otherwise the proof is finished We

consider 1198950= min119895 isin N | 119892

119895= 0 If 119868 is a first integral of

this system then there exists 119872 isin N such that 119868 = ℎ119872

+ sdot sdot sdot

and 0 = nabla119868 sdotF = 21199011199021198921198950ℎ1198950+1 + sdot sdot sdot Therefore 119892

1198950= 0 and this

is a contradiction

The next result shows that any integrable system (20)admits always a first integral of the form119867

1

Proposition 7 If system (20) is formally integrable then119867(119909 119910) = sum

119895ge0119891119895(119909)119910119895 with 119891

119895(119909) a formal function is a first

integral and moreover 1198910= sdot sdot sdot 119891

119901minus1= 0

Proof By applying the Poincare-Dulac normal form thereexists a change of variable to transform (20) into

(

V) = (

119901119906

minus119902V) + (

sum

119896ge1

119886119896119906ℎ119896

sum

119896ge1

119887119896Vℎ119896) (25)

where ℎ = 119906119902V119901 We can assume that the change of variable is

119906 = 119909 + sdot sdot sdot V = 119910(1 + sdot sdot sdot) because the axis 119910 = 0 and V = 0

are invariantConsidering the formal functions F(ℎ) = sum

119896ge1119888119896ℎ119896 and

G(ℎ) = sum119896ge1

119889119896ℎ119896 with 119888

119896= (119902119886119896minus 119901119887119896)2119901119902 and 119889

119896= (119902119886119896+

119901119887119896)2119901119902 we get

(

V) = (

119901119906

minus119902V) (1 +F (119906

119902V119901)) + (

119901119906

119902V) G (119906

119902V119901) (26)

whereF G are formal functions withF(0) = G(0) = 0Moreover by the scaling of the time 119889119905 = (1 +F)119889120591 we

can getF equiv 0 that is it is possible to transform (20) into

(1199061015840

V1015840) = (

119901119906

minus119902V) + (

119901119906

119902V)G (119906

119902V119901) (27)

whereG = G(1 +F) is a formal function withG(0) = 0If this system is formally integrable applying Lemma 6

we obtainG = 0Therefore119867(119909 119910) = (119909+sdot sdot sdot )119902119910119901(1+sdot sdot sdot )

119901=

sum119896ge119901

119891119896(119909)119910119896 is a first integral of system (20)

The main result of this work is as follows

Theorem 8 System (20) admits an analytic first integral119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896 if for each 119896 isin N such that 119896 = 119872119901

with119872 isin N is verified that the derivative (119872119902)119896

(0) = 0 where119896for 119896 ge 119901 is the analytical function

119896(119909) =

119892119896(119909)

119901 (1 + 1198750(119909)) 119891

119901(119909)119896

119891119901(119909) = exp(minusint

119909

0

(1199021198750(119904)

119901119904 (1 + 1198750(119904))

) 119889119904)

119892119901= 0 and for 119896 gt 119901

119892119896(119909) =

119896minus119901

sum

119895=1

((119896 minus 119895) 119891119896minus119895

(119909)119876119895+1

(119909) + 1198911015840

119896minus119895(119909) 119875119895(119909))

(28)

and for 119896 ge 119901

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

119904minus1minus(119896119902119901)

119896(119904) 119889119904) (29)

6 Abstract and Applied Analysis

Proof By Proposition 7 we have 1198910

= sdot sdot sdot = 119891119901minus1

= 0 Ifwe impose that 119867(119909 119910) be a first integral of system (20) weobtain that the first condition is

minus119901119902119891119901(119909) + 119901119909119891

1015840

119901(119909) (1 + 119875

0(119909)) = 0 (30)

The next coefficients for each power of 119910 are of the form

minus119902119896119891119896(119909) + 119901119909 (1 + 119875

0(119909)) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (31)

where 119892119901= 0 and 119892

119896(119909) for 119896 gt 119901 depends of the previous

119891119894(119909) for 119894 = 119901 119901 + 1 119896 minus 1 More specifically we obtain

the expression (28)The solution of homogeneous equation associated to (31)

is

119891ℎ

119896(119909) = exp(int

119896119902

119901119909 (1 + 1198750(119909))

) (32)

We can define the following analytical function 119887(119909) =

minus(119902119901119909)(1198750(119909)(1+119875

0(119909))) and the integrand of the previous

expression of 119891ℎ

119896is a rational function which admits the

following fraction decomposition

119896119902

119901119909 (1 + 1198750(119909))

=119896119902

119901119909+ 119896119887 (119909) (33)

Therefore 119891ℎ

119896(119909) = 119909

119896119902119901 exp(int 119896119887(119909)) =

119909119896119902119901

(exp(int1199090119887(119904)119889119904))

119896

and for the first equation whichcorresponds to 119896 = 119901 we have that 119892

119901(119909) = 0 Therefore

119891ℎ

119901(119909) has the form 119891

ℎ

119901(119909) = 119862

119901119909119902(119891119901(119909))119901

where 119891119901(119909) =

exp(int1199090119887(119904)119889119904) is an analytic function with 119891

119901(0) = 1 So

119891ℎ

119896(119909) = 119862

119896119909119896119902119901

119891119901(119909)119896 and the solution of (31) is given by

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

0

119904minus119896119902119901

119891119901(119904)minus119896

119901119904 (1 + 1198750(119904))

119892119896(119904) 119889119904)

(34)

Taking into account the form of 119892119896(119909) we obtain the

expression (29) for119891119896 Now it is straightforward to see that the

integral does not give logarithmic terms in the case 119896119902119901 notin N

and in the case 119896 = 119872119901 because (119872119902)

119896(0) = 0 If we choose

119862119896

= 0 we have that each 119891119896(119909) for all 119896 is an analytic

function and therefore system (20) under the assumptionsof the theorem admits a formal first integral of the form119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896

Corollary 9 System (20) admits a formal first integral of theform119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895 if one of the following conditions

holds

(a) 1198750(119909) equiv 0 and for each 119896 = 119872119901 119872 isin N 119892

119896(119909) is a

polynomial of degree at most119872119902 minus 1(b) 1198750(119909) equiv 119886119909

119903 with 119903 a positive integer and for each119896 = 119872119901 119872 isin N 119892

119896(119909) is a rational function of the

form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119902119903119901 where

119896is a

polynomial of degree at most119872119902 minus 1

Proof In case (a) it is enough to applyTheorem 8 for119891119901(119909) equiv

1 and 119896(119909) = 119892

119896(119909)119901 For 119896 = 119872119901 119872 isin N and we have

(119872119902)

119896(119909) = 119892

(119872119902)

119896119901 = 0 because the degree of 119892

119896(119909) is less

than 119902119872In case (b) we have that 119891

119901(119909) equiv (1 + 119886119909

119903)minus119902119903119901 and

119896(119909) = 119892

119896(119909)119901(1 + 119886119909

119903)1minus119902119903119901 Therefore

119896(119909) =

119896(119909)119901

and (119872119901)

119896(119909) = 0 because the degree of

119896(119909) is less than 119902119872

We obtain the result by applying also Theorem 8

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separate formin order to be applied directly to different examples

Lemma 10 If system (20)with 1198750(119909) equiv 0 admits a formal first

integral of the form119867(119909 119910) = sum119895ge119901

119891119895(119909)119910119895 then the following

conditions hold

(a) If for each 119896 such that 119896119902119901 notin N 119892119896is a polyno-

mial then it is possible to choose 119891119896polynomial with

deg(119891119896) = deg(119892

119896)

(b) If for each 119896 = 119872119901 119872 isin N 119892119896(119909) is a polynomial

with deg(119892119896) le 119872119902 minus 1 then it is possible to choose 119891

119896

polynomial with deg(119891119896) le 119872119902 minus 1

Proof By applying Theorem 8 we obtain 119891119901(119909) equiv 1 and

119896(119909) = 119892

119896(119909)119901 If we choose 119862

119896= 0 in (29) we obtain

119891119896(119909) = minus

1

119901119909119896119902119901

int

119909

119904minus1minus119896119902119901

119892119896(119904) 119889119904 (35)

In case (a) the result is trivial since the integral does not givelogarithmic terms because 119896119902119901 notin N

In case (b) 119896 = 119872119901 and 119891119896(119909) =

minus(1119901)119909119872119902

int119909

119904minus1minus119872119902

119892119896(119904) 119889119904Therefore we obtain the result

applying that 119892119896is a polynomial with deg(119892

119896) le 119872119902 minus 1

Based on the results presented in this work the followingproposition gives a large family of analytic systems that havean analytic first integral

Proposition 11 The analytic system

= 119901119909 + sum

119895ge1

119875119895(119909) 119910119895

= minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(36)

where 119875119895and 119876

119895are polynomials such that deg(119875

119895) le lfloor119895119902119901rfloor

deg(119876119895) le lfloor(119895 minus 1)119902119901rfloor minus 1 has an analytic first integral in

a neighborhood of the origin where lfloor119886rfloor is the integer part of119886 isin R

Proof From Proposition 7 if system (36) is integrable thena first integral is of the form 119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895

where 119891119895(119909) is a formal function Moreover 119875

0(119909) equiv 0 and

therefore by applying Theorem 8 we obtain 119891119901(119909) equiv 1

119896(119909) = 119892

119896(119909)119901 119892

119901= 0 and 119891

119901(119909) = 119909

119902

Abstract and Applied Analysis 7

We will prove now by induction that deg(119891119895) le lfloor119895119902119901rfloor

and deg(119892119895) le lfloor119895119902119901rfloor minus 1 for all 119895 isin N and 119901 le 119895

For 119895 = 119901 is true since 119891119901(119909) = 119909

119902 and 119892119901

= 0 If wesuppose that the hypothesis is true for 1 le 119895 le 119895

0minus1 we have

to prove that deg(1198921198950) le lfloor119895

0119902119901rfloor minus 1 and deg(119891

1198950) le lfloor119895

0119902119901rfloor

On the other hand applying (28) we have

deg (1198921198950)

le max1le119895le1198950minus119901

deg (1198911198950minus119895

)+deg (119876119895+1

) deg (11989110158401198950minus119895

) + deg (119875119895)

le lfloor1198950119902

119901rfloor minus 1

(37)

since for 1 le 119895 le 1198950minus 119901 we have

deg (1198911198950minus119895

) + deg (119876119895+1

) le lfloor(1198950minus 119895) 119902119901rfloor + lfloor

119895119902

119901rfloor minus 1

le lfloor1198950119902

119901rfloor minus 1

deg (11989110158401198950minus119895

) + deg (119875119895) le lfloor

(1198950minus 119895) 119902

119901rfloor minus 1 + lfloor

119895119902

119901rfloor

le lfloor1198950119902

119901rfloor minus 1

(38)

By applying Lemma 10 we have that deg(1198911198950) le lfloor119895

0119902119901rfloorminus

1 In particular deg(1198911198950) le lfloor119895

0119902119901rfloor

Therefore if 119896 = 119872119901 deg(119892119896) le 119872119902 minus 1 119892119872119902

119896(0) = 0 and

119872119902

119896(0) = 0To finish the proof it is enough to apply Theorem 8

However as it is difficult to determine the new coordi-nates where system (19) has 119876

0(119909) equiv 119876

1(119909) equiv 0 we are

going to work with the original system (5) imposing directlythese two conditions In order that one of the series 119867

1or

1198672be a formal first integral of system (5) we have that these

series must have the term 119909119902119910119901 as a first monomial in its

development We will see that in this case and under someconditionswe can use the inductionmethod in order to verifythe existence of a formal first integral

The following result is only established for the series 1198671

but we can obtain a similar result for the case where 1198672is a

formal first integral of system (5)

Proposition 12 We consider system (5)where wewrite1198651and

1198652into the form

1198651(119909 119910) =

infin

sum

119894+119895=2

119886119894119895119909119894119910119895

1198652(119909 119910) =

infin

sum

119894+119895=2

119887119894119895119909119894119910119895

(39)

where 119886119894119895and 119887119894119895are arbitrary constants If we impose that 119867

1

be a first integral of system (5) we obtain that the first conditionis

infin

sum

119894=2

1198871198940

1199091198941198911(119909) + 119901119909119891

1015840

0(119909) +

infin

sum

119894=2

11988611989401199091198941198911015840

0(119909) = 0 (40)

The proof of this proposition is also straightforwarddeveloping in powers of 119910 and taking the coefficient of thepower 119910

0 Hence in order to have a recursive differentialequations where each 119891

119894(119909) does not depend on 119891

119894+1(119909) we

must impose that 1198871198940

= 0 for all 119894 and consequently 1198910must

be a constant that we can take without loss of generality equalto zero The consequence as before is that if all 119887

1198940= 0 for all

119894 the system (5) has 119910 as invariant algebraic curveThe next coefficients for each power of 119910 are of the form

119896(minus119902 +

infin

sum

119894=1

1198871198941119909119894)119891119896(119909)

+ (119901119909 +

infin

sum

119894=2

1198861198940119909119894)1198911015840

119896(119909) + 119892

119896(119909) = 0

(41)

where 119892119896(119909) depends of the previous119891

119894(119909) for 119894 = 1 2 119896minus

1 and for the first equation which corresponds to 119896 = 1 wehave that 119892

1(119909) = 0 Therefore 119891

1(119909) has the form

1198911(119909) = exp(minusint

minus119902 + suminfin

119894=11198871198941119909119894

119901119909 + suminfin

119894=21198861198940119909119894

) (42)

The integrand of the previous expression of 1198911is a rational

function which admits the following fraction decomposition

minus119902 + suminfin

119894=11198871198941

119909119894

119901119909 + suminfin

119894=21198861198940

119909119894=

119860

119909+

119861 (119909)

119901 + suminfin

119894=21198861198940

119909119894minus1 (43)

where 119860 = minus119902119901 and 119861(119909) is a formal series Therefore1198911(119909) = 119909

1199021199011198911(119909) where 119891

1is the corresponding formal

series obtained after the integration Now we consider theparticular case 119901 = 1 The following theorem gives sufficientconditions to have formal integrability in this case A similarproposition can be established for the case when the firstintegral is of the form119867

2

Proposition 13 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if 119901 = 1

1198871198940= 0 for 119894 ge 2 and 119887

1198941= 0 for 119894 ge 1 and

119892119896(119909) =

119896(119909) 119875 (119909 0)

1199091198911(119909)minus119896

with (119896119902)

119896(0) = 0 (44)

Proof Under the assumptions of the theorem we have that1198910= 0 119891

1= 1199091199021198911(119909) and (41) takes the form

minus119896119902119891119896(119909) + 119875 (119909 0) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (45)

whose solution is given by

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909 119904minus119896119902

1198911(119904)minus119896

119875 (119904 0)119892119896(119904) 119889119904)

(46)

8 Abstract and Applied Analysis

Taking into account the form of 119892119896(119909) we obtain

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909

119904minus1minus119896119902

119896(119904) 119889119904) (47)

Now it is straightforward to see that the integral does not givelogarithmic terms because the 119896119902-th derivative of

119896(119904) at zero

is zero that is (119896119902)119896

(0) = 0 We have that each 119891119896(119909) for all

119896 is an analytic function and therefore system (5) under theassumptions of the theorem admits a formal first integral ofthe form (39)

Examples where Proposition 13 can be applied can befound in [7 12 16 17] where some partial results are givenWe also can establish the following corollary in the casewherethe 119892119896(119909) are polynomials or some irrational functions

Corollary 14 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if one of

the following conditions holds

(a) 119901 = 1 1198861198940

= 1198871198940

= 0 for 119894 ge 2 and 1198871198941

= 0 for 119894 ge 1 and119892119896(119909) is a polynomial of degree at most 119896119902 minus 1

(b) 119901 = 1 1198861198940= 0 for 119894 = 119903 + 1 119887

1198940= 0 for 119894 ge 2 and 119887

1198941= 0

for 119894 ge 1with 119903 a positive integer and 119892119896(119909) is a rational

function of the form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119896119902119903

where 119896is a polynomial of degree at most 119896119902 minus 1

Proof In case (a) for we have1198910= 0 and119891

1= 119909119902 (41) take the

form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119892

119896(119909) = 0 (48)

whose solution is given by

119891119896(119909) = 119862

119896119909119896119902

minus 119909119896119902

int

119909

119904minus1minus119896119902

119892119896(119904) 119889119904 (49)

and taking into account that 119892119896(119909) that depends on 119891

119894for 119894 =

1 2 119896 minus 1 is at most of degree 119896119902 minus 1 we obtain that 119891119896is

a polynomial Moreover in this case system (5) becomes

= 119901119909 + 1199101206011(119909 119910)

= minus119902119910 + 1199101206012(119909 119910)

(50)

where 1206011and 120601

2are analytic functions

In case (b) we have that 1198910= 0 119891

1= 119909119902(1 + 119886119909

119903)minus119902119903 and

(41) takes the form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119886119909

119903+11198911015840

119896(119909) + 119892

119896(119909) = 0 (51)

whose solution is given by

119891119896(119909) = 119909

119896119902(1 + 119886119909

119903)minus119896119902119903

times (119862119896minus int

119909

119904minus1minus119896119902

(1 + 119886119904)minus1+119896119902119903

119892119896(119904) 119889119904)

(52)

and taking into account the form of 119892119896(119909) we obtain that

119891119896(119909) is also of the form 119891

119896(119909) = 119901

119896(119909)(1 + 119886119909

119903)minus119896119902119904 where

119901119896(119909) is a polynomial because the integral in (52) does not

give logarithmic terms

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separateform in order to be applied directly to different examplesCorollary 14 generalizes a lot of cases obtained in the liter-ature when some concrete families of polynomials systemswere studiedWe have notmade an exhaustive study of all thecases included in these large families but for instance containsthe cases (3) and (4) of Theorem 3 in [17] and case (3) ofTheorem 2 in [16]

In resume when we study a family that satisfies thenecessary conditions we must impose if the family systemhas a formal first integral of the form 119867

1or 1198672 If none of

them work we can look for the proper coordinates (1199111 1199112)

defined by the stable and instable manifold of the 119901 minus119902

resonant singular point In this way we must study if at leastone of the separatrices of the resonant point is algebraic Thebest case is when both are algebraicTherefore the problem tofind proper coordinates (119911

1 1199112) becomes a problem of finding

the invariant algebraic curves of system (5) passing throughthe 119901 minus119902 resonant singular point The existence of invariantalgebraic curves takes a leading role in the formal integrabilitytheory and is the base to find the proper coordinates systemThis is shown in the examples provided in [12] that aresolved by ad hoc methods and where the changes of variablesproposed to solve the examples are always invariant algebraiccurves of the original system passing through the originIn fact in [12] it recalled the Abhyankar-Moch theorem[37] This theorem establishes assuming the existence of aninvariant rational algebraic curve passing through the originthe existence of a rational invertible change of variables suchthat the invariant curve becomes one of the invariant axesIn this paper this result is generalized in the sense that any119901 minus119902 resonant singular point is formal orbitally equivalentto a system with 119910 = 0 as invariant algebraic curve seeLemma 5

6 Examples

In this section we present some examples where the resultsdeveloped in this paper are applied

Example 15 We first consider the 1 minus1 resonant quadraticsystem given by

= 119909 + 119886201199092+ 11988611119909119910 + 119886

021199102

= minus119910 + 119887201199092+ 11988711119909119910 + 119887

021199102

(53)

The aim is not to do an exhaustive study of when system (53)has a formal first integral in a neighborhood of the originIn fact the complex center cases were studied by Dulac see[34] The aim is only to show how to use the results givenin this paper If we apply directly Corollary 14 statement (a)we obtain 119887

20= 0 (this condition implies that 119910 = 0 be an

algebraic invariant curve of system (53)) and 11988620

= 11988711

= 0 Inthis case the quadratic system (53) reads

= 119909 + 11988611119909119910 + 119886

021199102

= minus119910 + 119887021199102

(54)

Abstract and Applied Analysis 9

and admits a formal first integral and by Theorem 2 system(54) has an analytic first integral in a neighborhood of theorigin

In fact the associated equation to system (54) is a linearequation with respect to 119909 and the system has a Darboux firstintegral of the form

119867(119909 119910)

= (11988702119910 minus 1)

minus1minus(1198861111988702)

times [211988602

minus 2 (1198860211988611

+ 1198860211988702) 119910

+ (1198863

11minus 119886111198872

02) 119909119910 + (119886

021198862

11+ 119886021198861111988702) 1199102]

(55)

Example 16 Consider the 1 minus1 resonant LotkandashVolterraplanar complex quartic system given by

= 119909 (1 minus 11988610119909 minus 11988601119910 minus 119886201199092minus 11988611119909119910

minus119886021199102minus 119886301199093minus 119886211199092119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988710119909 minus 11988701119910 minus 119887201199092minus 11988711119909119910

minus119887021199102minus 119887301199093minus 119887211199092119910 minus 119887121199091199102minus 119887031199103)

(56)

The LotkandashVolterra planar complex systems have been stud-ied in several papers see [7 11 13ndash15] In any case the LotkandashVolterra planar complex quartic system is an open problemcomputationally infeasible However with the results of thispaperwe are going to see that we can give some sufficient con-ditions to have an analytic first integral without computingany resonant focus value We propose a formal first integralof the form 119867

1 that is of the form (16) The recurrence that

gives119891119894(119909) is given by minus119894119891

119894(119909)+119909119891

1015840

119894(119909)+119892

119894(119909) = 0where 119892

119894(119909)

is(119894 minus 3) 119887

03119891119894minus3

(119909) + (119894 minus 2) 11988702119891119894minus2

(119909) + (119894 minus 2) 11988712119909119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) + (119894 minus 1) 11988711119909119891119894minus1

(119909)

+ (119894 minus 1) 119887211199092119891119894minus1

(119909)

+ 11989411988710119909119891119894(119909) + 119894119887

201199092119891119894(119909)

+ 119894119887301199093119891119894(119909) minus 119886

031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909) minus 119886

1111990921198911015840

119894minus1(119909)

minus 1198861011990921198911015840

119894(119909) minus 119886

2011990931198911015840

119894(119909) minus 119886

3011990941198911015840

119894(119909)

(57)

Now we impose that 119892119894(119909) must be of degree lei minus1 This

implies that 11988610

= 11988620

= 11988630

= 11988710

= 11988720

= 11988730

= 0 and11988721

= 11988621

= 11988711

= 11988611

= 0 and in this case the recurrence isminus119894119891119894(119909) + 119909119891

1015840

119894(119909) + 119892

119894(119909) = 0 where 119892

119894(119909) is given by

(119894 minus 3) 11988703119891119894minus3

(119909) + (119894 minus 2) (11988702

+ 11988712119909)119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) minus 119886031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909)

(58)

We obtain that its solution is given by

119891119894(119909) = 119862

119894119909119894minus 119909119894int

119909

119904minus1minus119894

119892119894(119904) 119889119904 (59)

and taking into account that 119892119894(119909) is of degree le 119894 minus 1 we can

establish the following result

Proposition 17 The system

= 119909 (1 minus 11988601119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988701119910 minus 119887021199102minus 119887121199091199102minus 119887031199103)

(60)

has an analytic first integral in a neighborhood of the origin

In fact Proposition 17 is a consequence of Proposition 11This technique can be applied to several families of polyno-mial systems and in fact gives a new mechanism to obtainresonant centers and also real centers for the complex systemsthat have a real preimage (when this preimage exists see eg[18])

Acknowledgments

A Algaba and C Garcıa are supported by aMICINNFEDERGrant no MTM2010-20907-C02-02 and by the Consejerıade Educacıon y Ciencia de la Junta de Andalucıa (projectsEXC2008FQM-872 TIC-130 and FQM-276) J Gine is par-tially supported by a MICINNFEDER Grant no MTM2011-22877 and by Generalitat de Catalunya Grant no 2009SGR381

References

[1] AAlgaba E Gamero andCGarcıa ldquoThe integrability problemfor a class of planar systemsrdquo Nonlinearity vol 22 no 2 pp395ndash420 2009

[2] A Algaba E Freire E Gamero and C Garcıa ldquoMonodromycenter-focus and integrability problems for quasi-homogeneouspolynomial systemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1726ndash1736 2010

[3] A Algaba N Fuentes and C Garcıa ldquoCenters of quasi-homogeneous polynomial planar systemsrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 419ndash431 2012

[4] A Algaba C Garcıa and M Reyes ldquoExistence of an inverseintegrating factor center problem and integrability of a class ofnilpotent systemsrdquo Chaos Solitons amp Fractals vol 45 no 6 pp869ndash878 2012

[5] J Chavarriga H Giacomini J Gine and J Llibre ldquoOn theintegrability of two-dimensional flowsrdquo Journal of DifferentialEquations vol 157 no 1 pp 163ndash182 1999

[6] H Zoładek ldquoThe problem of center for resonant singular pointsof polynomial vector fieldsrdquo Journal of Differential Equationsvol 137 no 1 pp 94ndash118 1997

[7] X Chen J Gine V G Romanovski and D S Shafer ldquoThe1 -q resonant center problem for certain cubic Lotka-Volterrasystemsrdquo Applied Mathematics and Computation vol 218 no23 pp 11620ndash11633 2012

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Lemma 5 System (19) is orbitally equivalent to

= 119875 (119909 119910) = 119901119909 (1 + 1198750(119909)) + sum

119895ge1

119875119895(119909) 119910119895

= 119876 (119909 119910) = minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(20)

that is 1198760(119909) equiv 119876

1(119909) equiv 0 Moreover (119901 119902) are coprimes

Proof First we prove that we can achieve 1198760(119909) equiv 0 Assume

that 1198760(119909) equiv 0 and let 119873 = min119894 isin N cup 0 119887

1198940= 0 The

change of variable 119906 = 119909 V = 119910 minus (1198871198730

(119901(119873 + 1) + 119902))119909119873+1

transforms system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) V119895

V = 1199060(119906) minus 119902V (1 +

1(119906)) + sum

119895ge2

119895(119906) V119895

(21)

where 0(119906) = sum

119894ge01198871198940119906119894 and

1198940= 0 for 119894 = 0 1 119873

because V = minus ((119873 + 1)1198871198730

(119901(119873 + 1) + 119902))119906119873 hence

1199060(119906) = 119887

1198730119906119873+1

minus1199021198871198730

119901 (119873 + 1) + 119902119906119873+1

minus119901 (119873 + 1) 119887

1198730

119901 (119873 + 1) + 119902119906119873+1

+ O (119906119873+2

)

= O (119906119873+2

)

(22)

Therefore by means of successive change of variables we cantransform system (19) into

= 119901119906 (1 + 0(119906)) + sum

119895ge1

119895(119906) 119910119895

V = minus119902V (1 + 1(119906)) + sum

119895ge2

119895(119906) V119895

(23)

To complete the proof it is enough to apply the scaling 119889119905 =

119889119879(1 + 1(119906))

Lemma 6 Let G be a formal function with G(0) = 0 Thesystem

(

) = (

119901119909

minus119902119910) + (

119901119909

119902119910)G (119909

119902119910119901) (24)

is formally integrable if and only ifG equiv 0

Proof The sufficient condition is trivial since 119909119902119910119901 is a first

integral of ( )119879 = (119901119909 minus119902119910)119879

We now will prove the necessary condition Let F(119909 119910) =(119901119909 minus119902119910)

119879+ (119901119909 119902119910)

119879G(ℎ) where ℎ = 119909

119902119910119901 and G(ℎ) =

sum119895ge1

119892119895ℎ119895 G equiv 0 otherwise the proof is finished We

consider 1198950= min119895 isin N | 119892

119895= 0 If 119868 is a first integral of

this system then there exists 119872 isin N such that 119868 = ℎ119872

+ sdot sdot sdot

and 0 = nabla119868 sdotF = 21199011199021198921198950ℎ1198950+1 + sdot sdot sdot Therefore 119892

1198950= 0 and this

is a contradiction

The next result shows that any integrable system (20)admits always a first integral of the form119867

1

Proposition 7 If system (20) is formally integrable then119867(119909 119910) = sum

119895ge0119891119895(119909)119910119895 with 119891

119895(119909) a formal function is a first

integral and moreover 1198910= sdot sdot sdot 119891

119901minus1= 0

Proof By applying the Poincare-Dulac normal form thereexists a change of variable to transform (20) into

(

V) = (

119901119906

minus119902V) + (

sum

119896ge1

119886119896119906ℎ119896

sum

119896ge1

119887119896Vℎ119896) (25)

where ℎ = 119906119902V119901 We can assume that the change of variable is

119906 = 119909 + sdot sdot sdot V = 119910(1 + sdot sdot sdot) because the axis 119910 = 0 and V = 0

are invariantConsidering the formal functions F(ℎ) = sum

119896ge1119888119896ℎ119896 and

G(ℎ) = sum119896ge1

119889119896ℎ119896 with 119888

119896= (119902119886119896minus 119901119887119896)2119901119902 and 119889

119896= (119902119886119896+

119901119887119896)2119901119902 we get

(

V) = (

119901119906

minus119902V) (1 +F (119906

119902V119901)) + (

119901119906

119902V) G (119906

119902V119901) (26)

whereF G are formal functions withF(0) = G(0) = 0Moreover by the scaling of the time 119889119905 = (1 +F)119889120591 we

can getF equiv 0 that is it is possible to transform (20) into

(1199061015840

V1015840) = (

119901119906

minus119902V) + (

119901119906

119902V)G (119906

119902V119901) (27)

whereG = G(1 +F) is a formal function withG(0) = 0If this system is formally integrable applying Lemma 6

we obtainG = 0Therefore119867(119909 119910) = (119909+sdot sdot sdot )119902119910119901(1+sdot sdot sdot )

119901=

sum119896ge119901

119891119896(119909)119910119896 is a first integral of system (20)

The main result of this work is as follows

Theorem 8 System (20) admits an analytic first integral119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896 if for each 119896 isin N such that 119896 = 119872119901

with119872 isin N is verified that the derivative (119872119902)119896

(0) = 0 where119896for 119896 ge 119901 is the analytical function

119896(119909) =

119892119896(119909)

119901 (1 + 1198750(119909)) 119891

119901(119909)119896

119891119901(119909) = exp(minusint

119909

0

(1199021198750(119904)

119901119904 (1 + 1198750(119904))

) 119889119904)

119892119901= 0 and for 119896 gt 119901

119892119896(119909) =

119896minus119901

sum

119895=1

((119896 minus 119895) 119891119896minus119895

(119909)119876119895+1

(119909) + 1198911015840

119896minus119895(119909) 119875119895(119909))

(28)

and for 119896 ge 119901

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

119904minus1minus(119896119902119901)

119896(119904) 119889119904) (29)

6 Abstract and Applied Analysis

Proof By Proposition 7 we have 1198910

= sdot sdot sdot = 119891119901minus1

= 0 Ifwe impose that 119867(119909 119910) be a first integral of system (20) weobtain that the first condition is

minus119901119902119891119901(119909) + 119901119909119891

1015840

119901(119909) (1 + 119875

0(119909)) = 0 (30)

The next coefficients for each power of 119910 are of the form

minus119902119896119891119896(119909) + 119901119909 (1 + 119875

0(119909)) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (31)

where 119892119901= 0 and 119892

119896(119909) for 119896 gt 119901 depends of the previous

119891119894(119909) for 119894 = 119901 119901 + 1 119896 minus 1 More specifically we obtain

the expression (28)The solution of homogeneous equation associated to (31)

is

119891ℎ

119896(119909) = exp(int

119896119902

119901119909 (1 + 1198750(119909))

) (32)

We can define the following analytical function 119887(119909) =

minus(119902119901119909)(1198750(119909)(1+119875

0(119909))) and the integrand of the previous

expression of 119891ℎ

119896is a rational function which admits the

following fraction decomposition

119896119902

119901119909 (1 + 1198750(119909))

=119896119902

119901119909+ 119896119887 (119909) (33)

Therefore 119891ℎ

119896(119909) = 119909

119896119902119901 exp(int 119896119887(119909)) =

119909119896119902119901

(exp(int1199090119887(119904)119889119904))

119896

and for the first equation whichcorresponds to 119896 = 119901 we have that 119892

119901(119909) = 0 Therefore

119891ℎ

119901(119909) has the form 119891

ℎ

119901(119909) = 119862

119901119909119902(119891119901(119909))119901

where 119891119901(119909) =

exp(int1199090119887(119904)119889119904) is an analytic function with 119891

119901(0) = 1 So

119891ℎ

119896(119909) = 119862

119896119909119896119902119901

119891119901(119909)119896 and the solution of (31) is given by

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

0

119904minus119896119902119901

119891119901(119904)minus119896

119901119904 (1 + 1198750(119904))

119892119896(119904) 119889119904)

(34)

Taking into account the form of 119892119896(119909) we obtain the

expression (29) for119891119896 Now it is straightforward to see that the

integral does not give logarithmic terms in the case 119896119902119901 notin N

and in the case 119896 = 119872119901 because (119872119902)

119896(0) = 0 If we choose

119862119896

= 0 we have that each 119891119896(119909) for all 119896 is an analytic

function and therefore system (20) under the assumptionsof the theorem admits a formal first integral of the form119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896

Corollary 9 System (20) admits a formal first integral of theform119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895 if one of the following conditions

holds

(a) 1198750(119909) equiv 0 and for each 119896 = 119872119901 119872 isin N 119892

119896(119909) is a

polynomial of degree at most119872119902 minus 1(b) 1198750(119909) equiv 119886119909

119903 with 119903 a positive integer and for each119896 = 119872119901 119872 isin N 119892

119896(119909) is a rational function of the

form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119902119903119901 where

119896is a

polynomial of degree at most119872119902 minus 1

Proof In case (a) it is enough to applyTheorem 8 for119891119901(119909) equiv

1 and 119896(119909) = 119892

119896(119909)119901 For 119896 = 119872119901 119872 isin N and we have

(119872119902)

119896(119909) = 119892

(119872119902)

119896119901 = 0 because the degree of 119892

119896(119909) is less

than 119902119872In case (b) we have that 119891

119901(119909) equiv (1 + 119886119909

119903)minus119902119903119901 and

119896(119909) = 119892

119896(119909)119901(1 + 119886119909

119903)1minus119902119903119901 Therefore

119896(119909) =

119896(119909)119901

and (119872119901)

119896(119909) = 0 because the degree of

119896(119909) is less than 119902119872

We obtain the result by applying also Theorem 8

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separate formin order to be applied directly to different examples

Lemma 10 If system (20)with 1198750(119909) equiv 0 admits a formal first

integral of the form119867(119909 119910) = sum119895ge119901

119891119895(119909)119910119895 then the following

conditions hold

(a) If for each 119896 such that 119896119902119901 notin N 119892119896is a polyno-

mial then it is possible to choose 119891119896polynomial with

deg(119891119896) = deg(119892

119896)

(b) If for each 119896 = 119872119901 119872 isin N 119892119896(119909) is a polynomial

with deg(119892119896) le 119872119902 minus 1 then it is possible to choose 119891

119896

polynomial with deg(119891119896) le 119872119902 minus 1

Proof By applying Theorem 8 we obtain 119891119901(119909) equiv 1 and

119896(119909) = 119892

119896(119909)119901 If we choose 119862

119896= 0 in (29) we obtain

119891119896(119909) = minus

1

119901119909119896119902119901

int

119909

119904minus1minus119896119902119901

119892119896(119904) 119889119904 (35)

In case (a) the result is trivial since the integral does not givelogarithmic terms because 119896119902119901 notin N

In case (b) 119896 = 119872119901 and 119891119896(119909) =

minus(1119901)119909119872119902

int119909

119904minus1minus119872119902

119892119896(119904) 119889119904Therefore we obtain the result

applying that 119892119896is a polynomial with deg(119892

119896) le 119872119902 minus 1

Based on the results presented in this work the followingproposition gives a large family of analytic systems that havean analytic first integral

Proposition 11 The analytic system

= 119901119909 + sum

119895ge1

119875119895(119909) 119910119895

= minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(36)

where 119875119895and 119876

119895are polynomials such that deg(119875

119895) le lfloor119895119902119901rfloor

deg(119876119895) le lfloor(119895 minus 1)119902119901rfloor minus 1 has an analytic first integral in

a neighborhood of the origin where lfloor119886rfloor is the integer part of119886 isin R

Proof From Proposition 7 if system (36) is integrable thena first integral is of the form 119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895

where 119891119895(119909) is a formal function Moreover 119875

0(119909) equiv 0 and

therefore by applying Theorem 8 we obtain 119891119901(119909) equiv 1

119896(119909) = 119892

119896(119909)119901 119892

119901= 0 and 119891

119901(119909) = 119909

119902

Abstract and Applied Analysis 7

We will prove now by induction that deg(119891119895) le lfloor119895119902119901rfloor

and deg(119892119895) le lfloor119895119902119901rfloor minus 1 for all 119895 isin N and 119901 le 119895

For 119895 = 119901 is true since 119891119901(119909) = 119909

119902 and 119892119901

= 0 If wesuppose that the hypothesis is true for 1 le 119895 le 119895

0minus1 we have

to prove that deg(1198921198950) le lfloor119895

0119902119901rfloor minus 1 and deg(119891

1198950) le lfloor119895

0119902119901rfloor

On the other hand applying (28) we have

deg (1198921198950)

le max1le119895le1198950minus119901

deg (1198911198950minus119895

)+deg (119876119895+1

) deg (11989110158401198950minus119895

) + deg (119875119895)

le lfloor1198950119902

119901rfloor minus 1

(37)

since for 1 le 119895 le 1198950minus 119901 we have

deg (1198911198950minus119895

) + deg (119876119895+1

) le lfloor(1198950minus 119895) 119902119901rfloor + lfloor

119895119902

119901rfloor minus 1

le lfloor1198950119902

119901rfloor minus 1

deg (11989110158401198950minus119895

) + deg (119875119895) le lfloor

(1198950minus 119895) 119902

119901rfloor minus 1 + lfloor

119895119902

119901rfloor

le lfloor1198950119902

119901rfloor minus 1

(38)

By applying Lemma 10 we have that deg(1198911198950) le lfloor119895

0119902119901rfloorminus

1 In particular deg(1198911198950) le lfloor119895

0119902119901rfloor

Therefore if 119896 = 119872119901 deg(119892119896) le 119872119902 minus 1 119892119872119902

119896(0) = 0 and

119872119902

119896(0) = 0To finish the proof it is enough to apply Theorem 8

However as it is difficult to determine the new coordi-nates where system (19) has 119876

0(119909) equiv 119876

1(119909) equiv 0 we are

going to work with the original system (5) imposing directlythese two conditions In order that one of the series 119867

1or

1198672be a formal first integral of system (5) we have that these

series must have the term 119909119902119910119901 as a first monomial in its

development We will see that in this case and under someconditionswe can use the inductionmethod in order to verifythe existence of a formal first integral

The following result is only established for the series 1198671

but we can obtain a similar result for the case where 1198672is a

formal first integral of system (5)

Proposition 12 We consider system (5)where wewrite1198651and

1198652into the form

1198651(119909 119910) =

infin

sum

119894+119895=2

119886119894119895119909119894119910119895

1198652(119909 119910) =

infin

sum

119894+119895=2

119887119894119895119909119894119910119895

(39)

where 119886119894119895and 119887119894119895are arbitrary constants If we impose that 119867

1

be a first integral of system (5) we obtain that the first conditionis

infin

sum

119894=2

1198871198940

1199091198941198911(119909) + 119901119909119891

1015840

0(119909) +

infin

sum

119894=2

11988611989401199091198941198911015840

0(119909) = 0 (40)

The proof of this proposition is also straightforwarddeveloping in powers of 119910 and taking the coefficient of thepower 119910

0 Hence in order to have a recursive differentialequations where each 119891

119894(119909) does not depend on 119891

119894+1(119909) we

must impose that 1198871198940

= 0 for all 119894 and consequently 1198910must

be a constant that we can take without loss of generality equalto zero The consequence as before is that if all 119887

1198940= 0 for all

119894 the system (5) has 119910 as invariant algebraic curveThe next coefficients for each power of 119910 are of the form

119896(minus119902 +

infin

sum

119894=1

1198871198941119909119894)119891119896(119909)

+ (119901119909 +

infin

sum

119894=2

1198861198940119909119894)1198911015840

119896(119909) + 119892

119896(119909) = 0

(41)

where 119892119896(119909) depends of the previous119891

119894(119909) for 119894 = 1 2 119896minus

1 and for the first equation which corresponds to 119896 = 1 wehave that 119892

1(119909) = 0 Therefore 119891

1(119909) has the form

1198911(119909) = exp(minusint

minus119902 + suminfin

119894=11198871198941119909119894

119901119909 + suminfin

119894=21198861198940119909119894

) (42)

The integrand of the previous expression of 1198911is a rational

function which admits the following fraction decomposition

minus119902 + suminfin

119894=11198871198941

119909119894

119901119909 + suminfin

119894=21198861198940

119909119894=

119860

119909+

119861 (119909)

119901 + suminfin

119894=21198861198940

119909119894minus1 (43)

where 119860 = minus119902119901 and 119861(119909) is a formal series Therefore1198911(119909) = 119909

1199021199011198911(119909) where 119891

1is the corresponding formal

series obtained after the integration Now we consider theparticular case 119901 = 1 The following theorem gives sufficientconditions to have formal integrability in this case A similarproposition can be established for the case when the firstintegral is of the form119867

2

Proposition 13 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if 119901 = 1

1198871198940= 0 for 119894 ge 2 and 119887

1198941= 0 for 119894 ge 1 and

119892119896(119909) =

119896(119909) 119875 (119909 0)

1199091198911(119909)minus119896

with (119896119902)

119896(0) = 0 (44)

Proof Under the assumptions of the theorem we have that1198910= 0 119891

1= 1199091199021198911(119909) and (41) takes the form

minus119896119902119891119896(119909) + 119875 (119909 0) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (45)

whose solution is given by

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909 119904minus119896119902

1198911(119904)minus119896

119875 (119904 0)119892119896(119904) 119889119904)

(46)

8 Abstract and Applied Analysis

Taking into account the form of 119892119896(119909) we obtain

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909

119904minus1minus119896119902

119896(119904) 119889119904) (47)

Now it is straightforward to see that the integral does not givelogarithmic terms because the 119896119902-th derivative of

119896(119904) at zero

is zero that is (119896119902)119896

(0) = 0 We have that each 119891119896(119909) for all

119896 is an analytic function and therefore system (5) under theassumptions of the theorem admits a formal first integral ofthe form (39)

Examples where Proposition 13 can be applied can befound in [7 12 16 17] where some partial results are givenWe also can establish the following corollary in the casewherethe 119892119896(119909) are polynomials or some irrational functions

Corollary 14 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if one of

the following conditions holds

(a) 119901 = 1 1198861198940

= 1198871198940

= 0 for 119894 ge 2 and 1198871198941

= 0 for 119894 ge 1 and119892119896(119909) is a polynomial of degree at most 119896119902 minus 1

(b) 119901 = 1 1198861198940= 0 for 119894 = 119903 + 1 119887

1198940= 0 for 119894 ge 2 and 119887

1198941= 0

for 119894 ge 1with 119903 a positive integer and 119892119896(119909) is a rational

function of the form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119896119902119903

where 119896is a polynomial of degree at most 119896119902 minus 1

Proof In case (a) for we have1198910= 0 and119891

1= 119909119902 (41) take the

form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119892

119896(119909) = 0 (48)

whose solution is given by

119891119896(119909) = 119862

119896119909119896119902

minus 119909119896119902

int

119909

119904minus1minus119896119902

119892119896(119904) 119889119904 (49)

and taking into account that 119892119896(119909) that depends on 119891

119894for 119894 =

1 2 119896 minus 1 is at most of degree 119896119902 minus 1 we obtain that 119891119896is

a polynomial Moreover in this case system (5) becomes

= 119901119909 + 1199101206011(119909 119910)

= minus119902119910 + 1199101206012(119909 119910)

(50)

where 1206011and 120601

2are analytic functions

In case (b) we have that 1198910= 0 119891

1= 119909119902(1 + 119886119909

119903)minus119902119903 and

(41) takes the form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119886119909

119903+11198911015840

119896(119909) + 119892

119896(119909) = 0 (51)

whose solution is given by

119891119896(119909) = 119909

119896119902(1 + 119886119909

119903)minus119896119902119903

times (119862119896minus int

119909

119904minus1minus119896119902

(1 + 119886119904)minus1+119896119902119903

119892119896(119904) 119889119904)

(52)

and taking into account the form of 119892119896(119909) we obtain that

119891119896(119909) is also of the form 119891

119896(119909) = 119901

119896(119909)(1 + 119886119909

119903)minus119896119902119904 where

119901119896(119909) is a polynomial because the integral in (52) does not

give logarithmic terms

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separateform in order to be applied directly to different examplesCorollary 14 generalizes a lot of cases obtained in the liter-ature when some concrete families of polynomials systemswere studiedWe have notmade an exhaustive study of all thecases included in these large families but for instance containsthe cases (3) and (4) of Theorem 3 in [17] and case (3) ofTheorem 2 in [16]

In resume when we study a family that satisfies thenecessary conditions we must impose if the family systemhas a formal first integral of the form 119867

1or 1198672 If none of

them work we can look for the proper coordinates (1199111 1199112)

defined by the stable and instable manifold of the 119901 minus119902

resonant singular point In this way we must study if at leastone of the separatrices of the resonant point is algebraic Thebest case is when both are algebraicTherefore the problem tofind proper coordinates (119911

1 1199112) becomes a problem of finding

the invariant algebraic curves of system (5) passing throughthe 119901 minus119902 resonant singular point The existence of invariantalgebraic curves takes a leading role in the formal integrabilitytheory and is the base to find the proper coordinates systemThis is shown in the examples provided in [12] that aresolved by ad hoc methods and where the changes of variablesproposed to solve the examples are always invariant algebraiccurves of the original system passing through the originIn fact in [12] it recalled the Abhyankar-Moch theorem[37] This theorem establishes assuming the existence of aninvariant rational algebraic curve passing through the originthe existence of a rational invertible change of variables suchthat the invariant curve becomes one of the invariant axesIn this paper this result is generalized in the sense that any119901 minus119902 resonant singular point is formal orbitally equivalentto a system with 119910 = 0 as invariant algebraic curve seeLemma 5

6 Examples

In this section we present some examples where the resultsdeveloped in this paper are applied

Example 15 We first consider the 1 minus1 resonant quadraticsystem given by

= 119909 + 119886201199092+ 11988611119909119910 + 119886

021199102

= minus119910 + 119887201199092+ 11988711119909119910 + 119887

021199102

(53)

The aim is not to do an exhaustive study of when system (53)has a formal first integral in a neighborhood of the originIn fact the complex center cases were studied by Dulac see[34] The aim is only to show how to use the results givenin this paper If we apply directly Corollary 14 statement (a)we obtain 119887

20= 0 (this condition implies that 119910 = 0 be an

algebraic invariant curve of system (53)) and 11988620

= 11988711

= 0 Inthis case the quadratic system (53) reads

= 119909 + 11988611119909119910 + 119886

021199102

= minus119910 + 119887021199102

(54)

Abstract and Applied Analysis 9

and admits a formal first integral and by Theorem 2 system(54) has an analytic first integral in a neighborhood of theorigin

In fact the associated equation to system (54) is a linearequation with respect to 119909 and the system has a Darboux firstintegral of the form

119867(119909 119910)

= (11988702119910 minus 1)

minus1minus(1198861111988702)

times [211988602

minus 2 (1198860211988611

+ 1198860211988702) 119910

+ (1198863

11minus 119886111198872

02) 119909119910 + (119886

021198862

11+ 119886021198861111988702) 1199102]

(55)

Example 16 Consider the 1 minus1 resonant LotkandashVolterraplanar complex quartic system given by

= 119909 (1 minus 11988610119909 minus 11988601119910 minus 119886201199092minus 11988611119909119910

minus119886021199102minus 119886301199093minus 119886211199092119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988710119909 minus 11988701119910 minus 119887201199092minus 11988711119909119910

minus119887021199102minus 119887301199093minus 119887211199092119910 minus 119887121199091199102minus 119887031199103)

(56)

The LotkandashVolterra planar complex systems have been stud-ied in several papers see [7 11 13ndash15] In any case the LotkandashVolterra planar complex quartic system is an open problemcomputationally infeasible However with the results of thispaperwe are going to see that we can give some sufficient con-ditions to have an analytic first integral without computingany resonant focus value We propose a formal first integralof the form 119867

1 that is of the form (16) The recurrence that

gives119891119894(119909) is given by minus119894119891

119894(119909)+119909119891

1015840

119894(119909)+119892

119894(119909) = 0where 119892

119894(119909)

is(119894 minus 3) 119887

03119891119894minus3

(119909) + (119894 minus 2) 11988702119891119894minus2

(119909) + (119894 minus 2) 11988712119909119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) + (119894 minus 1) 11988711119909119891119894minus1

(119909)

+ (119894 minus 1) 119887211199092119891119894minus1

(119909)

+ 11989411988710119909119891119894(119909) + 119894119887

201199092119891119894(119909)

+ 119894119887301199093119891119894(119909) minus 119886

031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909) minus 119886

1111990921198911015840

119894minus1(119909)

minus 1198861011990921198911015840

119894(119909) minus 119886

2011990931198911015840

119894(119909) minus 119886

3011990941198911015840

119894(119909)

(57)

Now we impose that 119892119894(119909) must be of degree lei minus1 This

implies that 11988610

= 11988620

= 11988630

= 11988710

= 11988720

= 11988730

= 0 and11988721

= 11988621

= 11988711

= 11988611

= 0 and in this case the recurrence isminus119894119891119894(119909) + 119909119891

1015840

119894(119909) + 119892

119894(119909) = 0 where 119892

119894(119909) is given by

(119894 minus 3) 11988703119891119894minus3

(119909) + (119894 minus 2) (11988702

+ 11988712119909)119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) minus 119886031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909)

(58)

We obtain that its solution is given by

119891119894(119909) = 119862

119894119909119894minus 119909119894int

119909

119904minus1minus119894

119892119894(119904) 119889119904 (59)

and taking into account that 119892119894(119909) is of degree le 119894 minus 1 we can

establish the following result

Proposition 17 The system

= 119909 (1 minus 11988601119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988701119910 minus 119887021199102minus 119887121199091199102minus 119887031199103)

(60)

has an analytic first integral in a neighborhood of the origin

In fact Proposition 17 is a consequence of Proposition 11This technique can be applied to several families of polyno-mial systems and in fact gives a new mechanism to obtainresonant centers and also real centers for the complex systemsthat have a real preimage (when this preimage exists see eg[18])

Acknowledgments

A Algaba and C Garcıa are supported by aMICINNFEDERGrant no MTM2010-20907-C02-02 and by the Consejerıade Educacıon y Ciencia de la Junta de Andalucıa (projectsEXC2008FQM-872 TIC-130 and FQM-276) J Gine is par-tially supported by a MICINNFEDER Grant no MTM2011-22877 and by Generalitat de Catalunya Grant no 2009SGR381

References

[1] AAlgaba E Gamero andCGarcıa ldquoThe integrability problemfor a class of planar systemsrdquo Nonlinearity vol 22 no 2 pp395ndash420 2009

[2] A Algaba E Freire E Gamero and C Garcıa ldquoMonodromycenter-focus and integrability problems for quasi-homogeneouspolynomial systemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1726ndash1736 2010

[3] A Algaba N Fuentes and C Garcıa ldquoCenters of quasi-homogeneous polynomial planar systemsrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 419ndash431 2012

[4] A Algaba C Garcıa and M Reyes ldquoExistence of an inverseintegrating factor center problem and integrability of a class ofnilpotent systemsrdquo Chaos Solitons amp Fractals vol 45 no 6 pp869ndash878 2012

[5] J Chavarriga H Giacomini J Gine and J Llibre ldquoOn theintegrability of two-dimensional flowsrdquo Journal of DifferentialEquations vol 157 no 1 pp 163ndash182 1999

[6] H Zoładek ldquoThe problem of center for resonant singular pointsof polynomial vector fieldsrdquo Journal of Differential Equationsvol 137 no 1 pp 94ndash118 1997

[7] X Chen J Gine V G Romanovski and D S Shafer ldquoThe1 -q resonant center problem for certain cubic Lotka-Volterrasystemsrdquo Applied Mathematics and Computation vol 218 no23 pp 11620ndash11633 2012

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Proof By Proposition 7 we have 1198910

= sdot sdot sdot = 119891119901minus1

= 0 Ifwe impose that 119867(119909 119910) be a first integral of system (20) weobtain that the first condition is

minus119901119902119891119901(119909) + 119901119909119891

1015840

119901(119909) (1 + 119875

0(119909)) = 0 (30)

The next coefficients for each power of 119910 are of the form

minus119902119896119891119896(119909) + 119901119909 (1 + 119875

0(119909)) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (31)

where 119892119901= 0 and 119892

119896(119909) for 119896 gt 119901 depends of the previous

119891119894(119909) for 119894 = 119901 119901 + 1 119896 minus 1 More specifically we obtain

the expression (28)The solution of homogeneous equation associated to (31)

is

119891ℎ

119896(119909) = exp(int

119896119902

119901119909 (1 + 1198750(119909))

) (32)

We can define the following analytical function 119887(119909) =

minus(119902119901119909)(1198750(119909)(1+119875

0(119909))) and the integrand of the previous

expression of 119891ℎ

119896is a rational function which admits the

following fraction decomposition

119896119902

119901119909 (1 + 1198750(119909))

=119896119902

119901119909+ 119896119887 (119909) (33)

Therefore 119891ℎ

119896(119909) = 119909

119896119902119901 exp(int 119896119887(119909)) =

119909119896119902119901

(exp(int1199090119887(119904)119889119904))

119896

and for the first equation whichcorresponds to 119896 = 119901 we have that 119892

119901(119909) = 0 Therefore

119891ℎ

119901(119909) has the form 119891

ℎ

119901(119909) = 119862

119901119909119902(119891119901(119909))119901

where 119891119901(119909) =

exp(int1199090119887(119904)119889119904) is an analytic function with 119891

119901(0) = 1 So

119891ℎ

119896(119909) = 119862

119896119909119896119902119901

119891119901(119909)119896 and the solution of (31) is given by

119891119896(119909) = 119909

119896119902119901119891119901(119909)119896(119862119896minus int

119909

0

119904minus119896119902119901

119891119901(119904)minus119896

119901119904 (1 + 1198750(119904))

119892119896(119904) 119889119904)

(34)

Taking into account the form of 119892119896(119909) we obtain the

expression (29) for119891119896 Now it is straightforward to see that the

integral does not give logarithmic terms in the case 119896119902119901 notin N

and in the case 119896 = 119872119901 because (119872119902)

119896(0) = 0 If we choose

119862119896

= 0 we have that each 119891119896(119909) for all 119896 is an analytic

function and therefore system (20) under the assumptionsof the theorem admits a formal first integral of the form119867(119909 119910) = sum

119896ge119901119891119896(119909)119910119896

Corollary 9 System (20) admits a formal first integral of theform119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895 if one of the following conditions

holds

(a) 1198750(119909) equiv 0 and for each 119896 = 119872119901 119872 isin N 119892

119896(119909) is a

polynomial of degree at most119872119902 minus 1(b) 1198750(119909) equiv 119886119909

119903 with 119903 a positive integer and for each119896 = 119872119901 119872 isin N 119892

119896(119909) is a rational function of the

form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119902119903119901 where

119896is a

polynomial of degree at most119872119902 minus 1

Proof In case (a) it is enough to applyTheorem 8 for119891119901(119909) equiv

1 and 119896(119909) = 119892

119896(119909)119901 For 119896 = 119872119901 119872 isin N and we have

(119872119902)

119896(119909) = 119892

(119872119902)

119896119901 = 0 because the degree of 119892

119896(119909) is less

than 119902119872In case (b) we have that 119891

119901(119909) equiv (1 + 119886119909

119903)minus119902119903119901 and

119896(119909) = 119892

119896(119909)119901(1 + 119886119909

119903)1minus119902119903119901 Therefore

119896(119909) =

119896(119909)119901

and (119872119901)

119896(119909) = 0 because the degree of

119896(119909) is less than 119902119872

We obtain the result by applying also Theorem 8

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separate formin order to be applied directly to different examples

Lemma 10 If system (20)with 1198750(119909) equiv 0 admits a formal first

integral of the form119867(119909 119910) = sum119895ge119901

119891119895(119909)119910119895 then the following

conditions hold

(a) If for each 119896 such that 119896119902119901 notin N 119892119896is a polyno-

mial then it is possible to choose 119891119896polynomial with

deg(119891119896) = deg(119892

119896)

(b) If for each 119896 = 119872119901 119872 isin N 119892119896(119909) is a polynomial

with deg(119892119896) le 119872119902 minus 1 then it is possible to choose 119891

119896

polynomial with deg(119891119896) le 119872119902 minus 1

Proof By applying Theorem 8 we obtain 119891119901(119909) equiv 1 and

119896(119909) = 119892

119896(119909)119901 If we choose 119862

119896= 0 in (29) we obtain

119891119896(119909) = minus

1

119901119909119896119902119901

int

119909

119904minus1minus119896119902119901

119892119896(119904) 119889119904 (35)

In case (a) the result is trivial since the integral does not givelogarithmic terms because 119896119902119901 notin N

In case (b) 119896 = 119872119901 and 119891119896(119909) =

minus(1119901)119909119872119902

int119909

119904minus1minus119872119902

119892119896(119904) 119889119904Therefore we obtain the result

applying that 119892119896is a polynomial with deg(119892

119896) le 119872119902 minus 1

Based on the results presented in this work the followingproposition gives a large family of analytic systems that havean analytic first integral

Proposition 11 The analytic system

= 119901119909 + sum

119895ge1

119875119895(119909) 119910119895

= minus119902119910 + sum

119895ge2

119876119895(119909) 119910119895

(36)

where 119875119895and 119876

119895are polynomials such that deg(119875

119895) le lfloor119895119902119901rfloor

deg(119876119895) le lfloor(119895 minus 1)119902119901rfloor minus 1 has an analytic first integral in

a neighborhood of the origin where lfloor119886rfloor is the integer part of119886 isin R

Proof From Proposition 7 if system (36) is integrable thena first integral is of the form 119867(119909 119910) = sum

119895ge119901119891119895(119909)119910119895

where 119891119895(119909) is a formal function Moreover 119875

0(119909) equiv 0 and

therefore by applying Theorem 8 we obtain 119891119901(119909) equiv 1

119896(119909) = 119892

119896(119909)119901 119892

119901= 0 and 119891

119901(119909) = 119909

119902

Abstract and Applied Analysis 7

We will prove now by induction that deg(119891119895) le lfloor119895119902119901rfloor

and deg(119892119895) le lfloor119895119902119901rfloor minus 1 for all 119895 isin N and 119901 le 119895

For 119895 = 119901 is true since 119891119901(119909) = 119909

119902 and 119892119901

= 0 If wesuppose that the hypothesis is true for 1 le 119895 le 119895

0minus1 we have

to prove that deg(1198921198950) le lfloor119895

0119902119901rfloor minus 1 and deg(119891

1198950) le lfloor119895

0119902119901rfloor

On the other hand applying (28) we have

deg (1198921198950)

le max1le119895le1198950minus119901

deg (1198911198950minus119895

)+deg (119876119895+1

) deg (11989110158401198950minus119895

) + deg (119875119895)

le lfloor1198950119902

119901rfloor minus 1

(37)

since for 1 le 119895 le 1198950minus 119901 we have

deg (1198911198950minus119895

) + deg (119876119895+1

) le lfloor(1198950minus 119895) 119902119901rfloor + lfloor

119895119902

119901rfloor minus 1

le lfloor1198950119902

119901rfloor minus 1

deg (11989110158401198950minus119895

) + deg (119875119895) le lfloor

(1198950minus 119895) 119902

119901rfloor minus 1 + lfloor

119895119902

119901rfloor

le lfloor1198950119902

119901rfloor minus 1

(38)

By applying Lemma 10 we have that deg(1198911198950) le lfloor119895

0119902119901rfloorminus

1 In particular deg(1198911198950) le lfloor119895

0119902119901rfloor

Therefore if 119896 = 119872119901 deg(119892119896) le 119872119902 minus 1 119892119872119902

119896(0) = 0 and

119872119902

119896(0) = 0To finish the proof it is enough to apply Theorem 8

However as it is difficult to determine the new coordi-nates where system (19) has 119876

0(119909) equiv 119876

1(119909) equiv 0 we are

going to work with the original system (5) imposing directlythese two conditions In order that one of the series 119867

1or

1198672be a formal first integral of system (5) we have that these

series must have the term 119909119902119910119901 as a first monomial in its

development We will see that in this case and under someconditionswe can use the inductionmethod in order to verifythe existence of a formal first integral

The following result is only established for the series 1198671

but we can obtain a similar result for the case where 1198672is a

formal first integral of system (5)

Proposition 12 We consider system (5)where wewrite1198651and

1198652into the form

1198651(119909 119910) =

infin

sum

119894+119895=2

119886119894119895119909119894119910119895

1198652(119909 119910) =

infin

sum

119894+119895=2

119887119894119895119909119894119910119895

(39)

where 119886119894119895and 119887119894119895are arbitrary constants If we impose that 119867

1

be a first integral of system (5) we obtain that the first conditionis

infin

sum

119894=2

1198871198940

1199091198941198911(119909) + 119901119909119891

1015840

0(119909) +

infin

sum

119894=2

11988611989401199091198941198911015840

0(119909) = 0 (40)

The proof of this proposition is also straightforwarddeveloping in powers of 119910 and taking the coefficient of thepower 119910

0 Hence in order to have a recursive differentialequations where each 119891

119894(119909) does not depend on 119891

119894+1(119909) we

must impose that 1198871198940

= 0 for all 119894 and consequently 1198910must

be a constant that we can take without loss of generality equalto zero The consequence as before is that if all 119887

1198940= 0 for all

119894 the system (5) has 119910 as invariant algebraic curveThe next coefficients for each power of 119910 are of the form

119896(minus119902 +

infin

sum

119894=1

1198871198941119909119894)119891119896(119909)

+ (119901119909 +

infin

sum

119894=2

1198861198940119909119894)1198911015840

119896(119909) + 119892

119896(119909) = 0

(41)

where 119892119896(119909) depends of the previous119891

119894(119909) for 119894 = 1 2 119896minus

1 and for the first equation which corresponds to 119896 = 1 wehave that 119892

1(119909) = 0 Therefore 119891

1(119909) has the form

1198911(119909) = exp(minusint

minus119902 + suminfin

119894=11198871198941119909119894

119901119909 + suminfin

119894=21198861198940119909119894

) (42)

The integrand of the previous expression of 1198911is a rational

function which admits the following fraction decomposition

minus119902 + suminfin

119894=11198871198941

119909119894

119901119909 + suminfin

119894=21198861198940

119909119894=

119860

119909+

119861 (119909)

119901 + suminfin

119894=21198861198940

119909119894minus1 (43)

where 119860 = minus119902119901 and 119861(119909) is a formal series Therefore1198911(119909) = 119909

1199021199011198911(119909) where 119891

1is the corresponding formal

series obtained after the integration Now we consider theparticular case 119901 = 1 The following theorem gives sufficientconditions to have formal integrability in this case A similarproposition can be established for the case when the firstintegral is of the form119867

2

Proposition 13 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if 119901 = 1

1198871198940= 0 for 119894 ge 2 and 119887

1198941= 0 for 119894 ge 1 and

119892119896(119909) =

119896(119909) 119875 (119909 0)

1199091198911(119909)minus119896

with (119896119902)

119896(0) = 0 (44)

Proof Under the assumptions of the theorem we have that1198910= 0 119891

1= 1199091199021198911(119909) and (41) takes the form

minus119896119902119891119896(119909) + 119875 (119909 0) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (45)

whose solution is given by

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909 119904minus119896119902

1198911(119904)minus119896

119875 (119904 0)119892119896(119904) 119889119904)

(46)

8 Abstract and Applied Analysis

Taking into account the form of 119892119896(119909) we obtain

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909

119904minus1minus119896119902

119896(119904) 119889119904) (47)

Now it is straightforward to see that the integral does not givelogarithmic terms because the 119896119902-th derivative of

119896(119904) at zero

is zero that is (119896119902)119896

(0) = 0 We have that each 119891119896(119909) for all

119896 is an analytic function and therefore system (5) under theassumptions of the theorem admits a formal first integral ofthe form (39)

Examples where Proposition 13 can be applied can befound in [7 12 16 17] where some partial results are givenWe also can establish the following corollary in the casewherethe 119892119896(119909) are polynomials or some irrational functions

Corollary 14 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if one of

the following conditions holds

(a) 119901 = 1 1198861198940

= 1198871198940

= 0 for 119894 ge 2 and 1198871198941

= 0 for 119894 ge 1 and119892119896(119909) is a polynomial of degree at most 119896119902 minus 1

(b) 119901 = 1 1198861198940= 0 for 119894 = 119903 + 1 119887

1198940= 0 for 119894 ge 2 and 119887

1198941= 0

for 119894 ge 1with 119903 a positive integer and 119892119896(119909) is a rational

function of the form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119896119902119903

where 119896is a polynomial of degree at most 119896119902 minus 1

Proof In case (a) for we have1198910= 0 and119891

1= 119909119902 (41) take the

form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119892

119896(119909) = 0 (48)

whose solution is given by

119891119896(119909) = 119862

119896119909119896119902

minus 119909119896119902

int

119909

119904minus1minus119896119902

119892119896(119904) 119889119904 (49)

and taking into account that 119892119896(119909) that depends on 119891

119894for 119894 =

1 2 119896 minus 1 is at most of degree 119896119902 minus 1 we obtain that 119891119896is

a polynomial Moreover in this case system (5) becomes

= 119901119909 + 1199101206011(119909 119910)

= minus119902119910 + 1199101206012(119909 119910)

(50)

where 1206011and 120601

2are analytic functions

In case (b) we have that 1198910= 0 119891

1= 119909119902(1 + 119886119909

119903)minus119902119903 and

(41) takes the form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119886119909

119903+11198911015840

119896(119909) + 119892

119896(119909) = 0 (51)

whose solution is given by

119891119896(119909) = 119909

119896119902(1 + 119886119909

119903)minus119896119902119903

times (119862119896minus int

119909

119904minus1minus119896119902

(1 + 119886119904)minus1+119896119902119903

119892119896(119904) 119889119904)

(52)

and taking into account the form of 119892119896(119909) we obtain that

119891119896(119909) is also of the form 119891

119896(119909) = 119901

119896(119909)(1 + 119886119909

119903)minus119896119902119904 where

119901119896(119909) is a polynomial because the integral in (52) does not

give logarithmic terms

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separateform in order to be applied directly to different examplesCorollary 14 generalizes a lot of cases obtained in the liter-ature when some concrete families of polynomials systemswere studiedWe have notmade an exhaustive study of all thecases included in these large families but for instance containsthe cases (3) and (4) of Theorem 3 in [17] and case (3) ofTheorem 2 in [16]

In resume when we study a family that satisfies thenecessary conditions we must impose if the family systemhas a formal first integral of the form 119867

1or 1198672 If none of

them work we can look for the proper coordinates (1199111 1199112)

defined by the stable and instable manifold of the 119901 minus119902

resonant singular point In this way we must study if at leastone of the separatrices of the resonant point is algebraic Thebest case is when both are algebraicTherefore the problem tofind proper coordinates (119911

1 1199112) becomes a problem of finding

the invariant algebraic curves of system (5) passing throughthe 119901 minus119902 resonant singular point The existence of invariantalgebraic curves takes a leading role in the formal integrabilitytheory and is the base to find the proper coordinates systemThis is shown in the examples provided in [12] that aresolved by ad hoc methods and where the changes of variablesproposed to solve the examples are always invariant algebraiccurves of the original system passing through the originIn fact in [12] it recalled the Abhyankar-Moch theorem[37] This theorem establishes assuming the existence of aninvariant rational algebraic curve passing through the originthe existence of a rational invertible change of variables suchthat the invariant curve becomes one of the invariant axesIn this paper this result is generalized in the sense that any119901 minus119902 resonant singular point is formal orbitally equivalentto a system with 119910 = 0 as invariant algebraic curve seeLemma 5

6 Examples

In this section we present some examples where the resultsdeveloped in this paper are applied

Example 15 We first consider the 1 minus1 resonant quadraticsystem given by

= 119909 + 119886201199092+ 11988611119909119910 + 119886

021199102

= minus119910 + 119887201199092+ 11988711119909119910 + 119887

021199102

(53)

The aim is not to do an exhaustive study of when system (53)has a formal first integral in a neighborhood of the originIn fact the complex center cases were studied by Dulac see[34] The aim is only to show how to use the results givenin this paper If we apply directly Corollary 14 statement (a)we obtain 119887

20= 0 (this condition implies that 119910 = 0 be an

algebraic invariant curve of system (53)) and 11988620

= 11988711

= 0 Inthis case the quadratic system (53) reads

= 119909 + 11988611119909119910 + 119886

021199102

= minus119910 + 119887021199102

(54)

Abstract and Applied Analysis 9

and admits a formal first integral and by Theorem 2 system(54) has an analytic first integral in a neighborhood of theorigin

In fact the associated equation to system (54) is a linearequation with respect to 119909 and the system has a Darboux firstintegral of the form

119867(119909 119910)

= (11988702119910 minus 1)

minus1minus(1198861111988702)

times [211988602

minus 2 (1198860211988611

+ 1198860211988702) 119910

+ (1198863

11minus 119886111198872

02) 119909119910 + (119886

021198862

11+ 119886021198861111988702) 1199102]

(55)

Example 16 Consider the 1 minus1 resonant LotkandashVolterraplanar complex quartic system given by

= 119909 (1 minus 11988610119909 minus 11988601119910 minus 119886201199092minus 11988611119909119910

minus119886021199102minus 119886301199093minus 119886211199092119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988710119909 minus 11988701119910 minus 119887201199092minus 11988711119909119910

minus119887021199102minus 119887301199093minus 119887211199092119910 minus 119887121199091199102minus 119887031199103)

(56)

The LotkandashVolterra planar complex systems have been stud-ied in several papers see [7 11 13ndash15] In any case the LotkandashVolterra planar complex quartic system is an open problemcomputationally infeasible However with the results of thispaperwe are going to see that we can give some sufficient con-ditions to have an analytic first integral without computingany resonant focus value We propose a formal first integralof the form 119867

1 that is of the form (16) The recurrence that

gives119891119894(119909) is given by minus119894119891

119894(119909)+119909119891

1015840

119894(119909)+119892

119894(119909) = 0where 119892

119894(119909)

is(119894 minus 3) 119887

03119891119894minus3

(119909) + (119894 minus 2) 11988702119891119894minus2

(119909) + (119894 minus 2) 11988712119909119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) + (119894 minus 1) 11988711119909119891119894minus1

(119909)

+ (119894 minus 1) 119887211199092119891119894minus1

(119909)

+ 11989411988710119909119891119894(119909) + 119894119887

201199092119891119894(119909)

+ 119894119887301199093119891119894(119909) minus 119886

031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909) minus 119886

1111990921198911015840

119894minus1(119909)

minus 1198861011990921198911015840

119894(119909) minus 119886

2011990931198911015840

119894(119909) minus 119886

3011990941198911015840

119894(119909)

(57)

Now we impose that 119892119894(119909) must be of degree lei minus1 This

implies that 11988610

= 11988620

= 11988630

= 11988710

= 11988720

= 11988730

= 0 and11988721

= 11988621

= 11988711

= 11988611

= 0 and in this case the recurrence isminus119894119891119894(119909) + 119909119891

1015840

119894(119909) + 119892

119894(119909) = 0 where 119892

119894(119909) is given by

(119894 minus 3) 11988703119891119894minus3

(119909) + (119894 minus 2) (11988702

+ 11988712119909)119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) minus 119886031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909)

(58)

We obtain that its solution is given by

119891119894(119909) = 119862

119894119909119894minus 119909119894int

119909

119904minus1minus119894

119892119894(119904) 119889119904 (59)

and taking into account that 119892119894(119909) is of degree le 119894 minus 1 we can

establish the following result

Proposition 17 The system

= 119909 (1 minus 11988601119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988701119910 minus 119887021199102minus 119887121199091199102minus 119887031199103)

(60)

has an analytic first integral in a neighborhood of the origin

In fact Proposition 17 is a consequence of Proposition 11This technique can be applied to several families of polyno-mial systems and in fact gives a new mechanism to obtainresonant centers and also real centers for the complex systemsthat have a real preimage (when this preimage exists see eg[18])

Acknowledgments

A Algaba and C Garcıa are supported by aMICINNFEDERGrant no MTM2010-20907-C02-02 and by the Consejerıade Educacıon y Ciencia de la Junta de Andalucıa (projectsEXC2008FQM-872 TIC-130 and FQM-276) J Gine is par-tially supported by a MICINNFEDER Grant no MTM2011-22877 and by Generalitat de Catalunya Grant no 2009SGR381

References

[1] AAlgaba E Gamero andCGarcıa ldquoThe integrability problemfor a class of planar systemsrdquo Nonlinearity vol 22 no 2 pp395ndash420 2009

[2] A Algaba E Freire E Gamero and C Garcıa ldquoMonodromycenter-focus and integrability problems for quasi-homogeneouspolynomial systemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1726ndash1736 2010

[3] A Algaba N Fuentes and C Garcıa ldquoCenters of quasi-homogeneous polynomial planar systemsrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 419ndash431 2012

[4] A Algaba C Garcıa and M Reyes ldquoExistence of an inverseintegrating factor center problem and integrability of a class ofnilpotent systemsrdquo Chaos Solitons amp Fractals vol 45 no 6 pp869ndash878 2012

[5] J Chavarriga H Giacomini J Gine and J Llibre ldquoOn theintegrability of two-dimensional flowsrdquo Journal of DifferentialEquations vol 157 no 1 pp 163ndash182 1999

[6] H Zoładek ldquoThe problem of center for resonant singular pointsof polynomial vector fieldsrdquo Journal of Differential Equationsvol 137 no 1 pp 94ndash118 1997

[7] X Chen J Gine V G Romanovski and D S Shafer ldquoThe1 -q resonant center problem for certain cubic Lotka-Volterrasystemsrdquo Applied Mathematics and Computation vol 218 no23 pp 11620ndash11633 2012

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

We will prove now by induction that deg(119891119895) le lfloor119895119902119901rfloor

and deg(119892119895) le lfloor119895119902119901rfloor minus 1 for all 119895 isin N and 119901 le 119895

For 119895 = 119901 is true since 119891119901(119909) = 119909

119902 and 119892119901

= 0 If wesuppose that the hypothesis is true for 1 le 119895 le 119895

0minus1 we have

to prove that deg(1198921198950) le lfloor119895

0119902119901rfloor minus 1 and deg(119891

1198950) le lfloor119895

0119902119901rfloor

On the other hand applying (28) we have

deg (1198921198950)

le max1le119895le1198950minus119901

deg (1198911198950minus119895

)+deg (119876119895+1

) deg (11989110158401198950minus119895

) + deg (119875119895)

le lfloor1198950119902

119901rfloor minus 1

(37)

since for 1 le 119895 le 1198950minus 119901 we have

deg (1198911198950minus119895

) + deg (119876119895+1

) le lfloor(1198950minus 119895) 119902119901rfloor + lfloor

119895119902

119901rfloor minus 1

le lfloor1198950119902

119901rfloor minus 1

deg (11989110158401198950minus119895

) + deg (119875119895) le lfloor

(1198950minus 119895) 119902

119901rfloor minus 1 + lfloor

119895119902

119901rfloor

le lfloor1198950119902

119901rfloor minus 1

(38)

By applying Lemma 10 we have that deg(1198911198950) le lfloor119895

0119902119901rfloorminus

1 In particular deg(1198911198950) le lfloor119895

0119902119901rfloor

Therefore if 119896 = 119872119901 deg(119892119896) le 119872119902 minus 1 119892119872119902

119896(0) = 0 and

119872119902

119896(0) = 0To finish the proof it is enough to apply Theorem 8

However as it is difficult to determine the new coordi-nates where system (19) has 119876

0(119909) equiv 119876

1(119909) equiv 0 we are

going to work with the original system (5) imposing directlythese two conditions In order that one of the series 119867

1or

1198672be a formal first integral of system (5) we have that these

series must have the term 119909119902119910119901 as a first monomial in its

development We will see that in this case and under someconditionswe can use the inductionmethod in order to verifythe existence of a formal first integral

The following result is only established for the series 1198671

but we can obtain a similar result for the case where 1198672is a

formal first integral of system (5)

Proposition 12 We consider system (5)where wewrite1198651and

1198652into the form

1198651(119909 119910) =

infin

sum

119894+119895=2

119886119894119895119909119894119910119895

1198652(119909 119910) =

infin

sum

119894+119895=2

119887119894119895119909119894119910119895

(39)

where 119886119894119895and 119887119894119895are arbitrary constants If we impose that 119867

1

be a first integral of system (5) we obtain that the first conditionis

infin

sum

119894=2

1198871198940

1199091198941198911(119909) + 119901119909119891

1015840

0(119909) +

infin

sum

119894=2

11988611989401199091198941198911015840

0(119909) = 0 (40)

The proof of this proposition is also straightforwarddeveloping in powers of 119910 and taking the coefficient of thepower 119910

0 Hence in order to have a recursive differentialequations where each 119891

119894(119909) does not depend on 119891

119894+1(119909) we

must impose that 1198871198940

= 0 for all 119894 and consequently 1198910must

be a constant that we can take without loss of generality equalto zero The consequence as before is that if all 119887

1198940= 0 for all

119894 the system (5) has 119910 as invariant algebraic curveThe next coefficients for each power of 119910 are of the form

119896(minus119902 +

infin

sum

119894=1

1198871198941119909119894)119891119896(119909)

+ (119901119909 +

infin

sum

119894=2

1198861198940119909119894)1198911015840

119896(119909) + 119892

119896(119909) = 0

(41)

where 119892119896(119909) depends of the previous119891

119894(119909) for 119894 = 1 2 119896minus

1 and for the first equation which corresponds to 119896 = 1 wehave that 119892

1(119909) = 0 Therefore 119891

1(119909) has the form

1198911(119909) = exp(minusint

minus119902 + suminfin

119894=11198871198941119909119894

119901119909 + suminfin

119894=21198861198940119909119894

) (42)

The integrand of the previous expression of 1198911is a rational

function which admits the following fraction decomposition

minus119902 + suminfin

119894=11198871198941

119909119894

119901119909 + suminfin

119894=21198861198940

119909119894=

119860

119909+

119861 (119909)

119901 + suminfin

119894=21198861198940

119909119894minus1 (43)

where 119860 = minus119902119901 and 119861(119909) is a formal series Therefore1198911(119909) = 119909

1199021199011198911(119909) where 119891

1is the corresponding formal

series obtained after the integration Now we consider theparticular case 119901 = 1 The following theorem gives sufficientconditions to have formal integrability in this case A similarproposition can be established for the case when the firstintegral is of the form119867

2

Proposition 13 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if 119901 = 1

1198871198940= 0 for 119894 ge 2 and 119887

1198941= 0 for 119894 ge 1 and

119892119896(119909) =

119896(119909) 119875 (119909 0)

1199091198911(119909)minus119896

with (119896119902)

119896(0) = 0 (44)

Proof Under the assumptions of the theorem we have that1198910= 0 119891

1= 1199091199021198911(119909) and (41) takes the form

minus119896119902119891119896(119909) + 119875 (119909 0) 119891

1015840

119896(119909) + 119892

119896(119909) = 0 (45)

whose solution is given by

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909 119904minus119896119902

1198911(119904)minus119896

119875 (119904 0)119892119896(119904) 119889119904)

(46)

8 Abstract and Applied Analysis

Taking into account the form of 119892119896(119909) we obtain

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909

119904minus1minus119896119902

119896(119904) 119889119904) (47)

Now it is straightforward to see that the integral does not givelogarithmic terms because the 119896119902-th derivative of

119896(119904) at zero

is zero that is (119896119902)119896

(0) = 0 We have that each 119891119896(119909) for all

119896 is an analytic function and therefore system (5) under theassumptions of the theorem admits a formal first integral ofthe form (39)

Examples where Proposition 13 can be applied can befound in [7 12 16 17] where some partial results are givenWe also can establish the following corollary in the casewherethe 119892119896(119909) are polynomials or some irrational functions

Corollary 14 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if one of

the following conditions holds

(a) 119901 = 1 1198861198940

= 1198871198940

= 0 for 119894 ge 2 and 1198871198941

= 0 for 119894 ge 1 and119892119896(119909) is a polynomial of degree at most 119896119902 minus 1

(b) 119901 = 1 1198861198940= 0 for 119894 = 119903 + 1 119887

1198940= 0 for 119894 ge 2 and 119887

1198941= 0

for 119894 ge 1with 119903 a positive integer and 119892119896(119909) is a rational

function of the form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119896119902119903

where 119896is a polynomial of degree at most 119896119902 minus 1

Proof In case (a) for we have1198910= 0 and119891

1= 119909119902 (41) take the

form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119892

119896(119909) = 0 (48)

whose solution is given by

119891119896(119909) = 119862

119896119909119896119902

minus 119909119896119902

int

119909

119904minus1minus119896119902

119892119896(119904) 119889119904 (49)

and taking into account that 119892119896(119909) that depends on 119891

119894for 119894 =

1 2 119896 minus 1 is at most of degree 119896119902 minus 1 we obtain that 119891119896is

a polynomial Moreover in this case system (5) becomes

= 119901119909 + 1199101206011(119909 119910)

= minus119902119910 + 1199101206012(119909 119910)

(50)

where 1206011and 120601

2are analytic functions

In case (b) we have that 1198910= 0 119891

1= 119909119902(1 + 119886119909

119903)minus119902119903 and

(41) takes the form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119886119909

119903+11198911015840

119896(119909) + 119892

119896(119909) = 0 (51)

whose solution is given by

119891119896(119909) = 119909

119896119902(1 + 119886119909

119903)minus119896119902119903

times (119862119896minus int

119909

119904minus1minus119896119902

(1 + 119886119904)minus1+119896119902119903

119892119896(119904) 119889119904)

(52)

and taking into account the form of 119892119896(119909) we obtain that

119891119896(119909) is also of the form 119891

119896(119909) = 119901

119896(119909)(1 + 119886119909

119903)minus119896119902119904 where

119901119896(119909) is a polynomial because the integral in (52) does not

give logarithmic terms

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separateform in order to be applied directly to different examplesCorollary 14 generalizes a lot of cases obtained in the liter-ature when some concrete families of polynomials systemswere studiedWe have notmade an exhaustive study of all thecases included in these large families but for instance containsthe cases (3) and (4) of Theorem 3 in [17] and case (3) ofTheorem 2 in [16]

In resume when we study a family that satisfies thenecessary conditions we must impose if the family systemhas a formal first integral of the form 119867

1or 1198672 If none of

them work we can look for the proper coordinates (1199111 1199112)

defined by the stable and instable manifold of the 119901 minus119902

resonant singular point In this way we must study if at leastone of the separatrices of the resonant point is algebraic Thebest case is when both are algebraicTherefore the problem tofind proper coordinates (119911

1 1199112) becomes a problem of finding

the invariant algebraic curves of system (5) passing throughthe 119901 minus119902 resonant singular point The existence of invariantalgebraic curves takes a leading role in the formal integrabilitytheory and is the base to find the proper coordinates systemThis is shown in the examples provided in [12] that aresolved by ad hoc methods and where the changes of variablesproposed to solve the examples are always invariant algebraiccurves of the original system passing through the originIn fact in [12] it recalled the Abhyankar-Moch theorem[37] This theorem establishes assuming the existence of aninvariant rational algebraic curve passing through the originthe existence of a rational invertible change of variables suchthat the invariant curve becomes one of the invariant axesIn this paper this result is generalized in the sense that any119901 minus119902 resonant singular point is formal orbitally equivalentto a system with 119910 = 0 as invariant algebraic curve seeLemma 5

6 Examples

In this section we present some examples where the resultsdeveloped in this paper are applied

Example 15 We first consider the 1 minus1 resonant quadraticsystem given by

= 119909 + 119886201199092+ 11988611119909119910 + 119886

021199102

= minus119910 + 119887201199092+ 11988711119909119910 + 119887

021199102

(53)

The aim is not to do an exhaustive study of when system (53)has a formal first integral in a neighborhood of the originIn fact the complex center cases were studied by Dulac see[34] The aim is only to show how to use the results givenin this paper If we apply directly Corollary 14 statement (a)we obtain 119887

20= 0 (this condition implies that 119910 = 0 be an

algebraic invariant curve of system (53)) and 11988620

= 11988711

= 0 Inthis case the quadratic system (53) reads

= 119909 + 11988611119909119910 + 119886

021199102

= minus119910 + 119887021199102

(54)

Abstract and Applied Analysis 9

and admits a formal first integral and by Theorem 2 system(54) has an analytic first integral in a neighborhood of theorigin

In fact the associated equation to system (54) is a linearequation with respect to 119909 and the system has a Darboux firstintegral of the form

119867(119909 119910)

= (11988702119910 minus 1)

minus1minus(1198861111988702)

times [211988602

minus 2 (1198860211988611

+ 1198860211988702) 119910

+ (1198863

11minus 119886111198872

02) 119909119910 + (119886

021198862

11+ 119886021198861111988702) 1199102]

(55)

Example 16 Consider the 1 minus1 resonant LotkandashVolterraplanar complex quartic system given by

= 119909 (1 minus 11988610119909 minus 11988601119910 minus 119886201199092minus 11988611119909119910

minus119886021199102minus 119886301199093minus 119886211199092119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988710119909 minus 11988701119910 minus 119887201199092minus 11988711119909119910

minus119887021199102minus 119887301199093minus 119887211199092119910 minus 119887121199091199102minus 119887031199103)

(56)

The LotkandashVolterra planar complex systems have been stud-ied in several papers see [7 11 13ndash15] In any case the LotkandashVolterra planar complex quartic system is an open problemcomputationally infeasible However with the results of thispaperwe are going to see that we can give some sufficient con-ditions to have an analytic first integral without computingany resonant focus value We propose a formal first integralof the form 119867

1 that is of the form (16) The recurrence that

gives119891119894(119909) is given by minus119894119891

119894(119909)+119909119891

1015840

119894(119909)+119892

119894(119909) = 0where 119892

119894(119909)

is(119894 minus 3) 119887

03119891119894minus3

(119909) + (119894 minus 2) 11988702119891119894minus2

(119909) + (119894 minus 2) 11988712119909119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) + (119894 minus 1) 11988711119909119891119894minus1

(119909)

+ (119894 minus 1) 119887211199092119891119894minus1

(119909)

+ 11989411988710119909119891119894(119909) + 119894119887

201199092119891119894(119909)

+ 119894119887301199093119891119894(119909) minus 119886

031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909) minus 119886

1111990921198911015840

119894minus1(119909)

minus 1198861011990921198911015840

119894(119909) minus 119886

2011990931198911015840

119894(119909) minus 119886

3011990941198911015840

119894(119909)

(57)

Now we impose that 119892119894(119909) must be of degree lei minus1 This

implies that 11988610

= 11988620

= 11988630

= 11988710

= 11988720

= 11988730

= 0 and11988721

= 11988621

= 11988711

= 11988611

= 0 and in this case the recurrence isminus119894119891119894(119909) + 119909119891

1015840

119894(119909) + 119892

119894(119909) = 0 where 119892

119894(119909) is given by

(119894 minus 3) 11988703119891119894minus3

(119909) + (119894 minus 2) (11988702

+ 11988712119909)119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) minus 119886031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909)

(58)

We obtain that its solution is given by

119891119894(119909) = 119862

119894119909119894minus 119909119894int

119909

119904minus1minus119894

119892119894(119904) 119889119904 (59)

and taking into account that 119892119894(119909) is of degree le 119894 minus 1 we can

establish the following result

Proposition 17 The system

= 119909 (1 minus 11988601119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988701119910 minus 119887021199102minus 119887121199091199102minus 119887031199103)

(60)

has an analytic first integral in a neighborhood of the origin

In fact Proposition 17 is a consequence of Proposition 11This technique can be applied to several families of polyno-mial systems and in fact gives a new mechanism to obtainresonant centers and also real centers for the complex systemsthat have a real preimage (when this preimage exists see eg[18])

Acknowledgments

A Algaba and C Garcıa are supported by aMICINNFEDERGrant no MTM2010-20907-C02-02 and by the Consejerıade Educacıon y Ciencia de la Junta de Andalucıa (projectsEXC2008FQM-872 TIC-130 and FQM-276) J Gine is par-tially supported by a MICINNFEDER Grant no MTM2011-22877 and by Generalitat de Catalunya Grant no 2009SGR381

References

[1] AAlgaba E Gamero andCGarcıa ldquoThe integrability problemfor a class of planar systemsrdquo Nonlinearity vol 22 no 2 pp395ndash420 2009

[2] A Algaba E Freire E Gamero and C Garcıa ldquoMonodromycenter-focus and integrability problems for quasi-homogeneouspolynomial systemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1726ndash1736 2010

[3] A Algaba N Fuentes and C Garcıa ldquoCenters of quasi-homogeneous polynomial planar systemsrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 419ndash431 2012

[4] A Algaba C Garcıa and M Reyes ldquoExistence of an inverseintegrating factor center problem and integrability of a class ofnilpotent systemsrdquo Chaos Solitons amp Fractals vol 45 no 6 pp869ndash878 2012

[5] J Chavarriga H Giacomini J Gine and J Llibre ldquoOn theintegrability of two-dimensional flowsrdquo Journal of DifferentialEquations vol 157 no 1 pp 163ndash182 1999

[6] H Zoładek ldquoThe problem of center for resonant singular pointsof polynomial vector fieldsrdquo Journal of Differential Equationsvol 137 no 1 pp 94ndash118 1997

[7] X Chen J Gine V G Romanovski and D S Shafer ldquoThe1 -q resonant center problem for certain cubic Lotka-Volterrasystemsrdquo Applied Mathematics and Computation vol 218 no23 pp 11620ndash11633 2012

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

Taking into account the form of 119892119896(119909) we obtain

119891119896(119909) = 119909

1198961199021198911(119909)119896(119862119896minus int

119909

119904minus1minus119896119902

119896(119904) 119889119904) (47)

Now it is straightforward to see that the integral does not givelogarithmic terms because the 119896119902-th derivative of

119896(119904) at zero

is zero that is (119896119902)119896

(0) = 0 We have that each 119891119896(119909) for all

119896 is an analytic function and therefore system (5) under theassumptions of the theorem admits a formal first integral ofthe form (39)

Examples where Proposition 13 can be applied can befound in [7 12 16 17] where some partial results are givenWe also can establish the following corollary in the casewherethe 119892119896(119909) are polynomials or some irrational functions

Corollary 14 System (5) where we write 1198651and 119865

2into the

form (39) admits a formal first integral of the form1198671if one of

the following conditions holds

(a) 119901 = 1 1198861198940

= 1198871198940

= 0 for 119894 ge 2 and 1198871198941

= 0 for 119894 ge 1 and119892119896(119909) is a polynomial of degree at most 119896119902 minus 1

(b) 119901 = 1 1198861198940= 0 for 119894 = 119903 + 1 119887

1198940= 0 for 119894 ge 2 and 119887

1198941= 0

for 119894 ge 1with 119903 a positive integer and 119892119896(119909) is a rational

function of the form 119892119896(119909) =

119896(119909)(1 + 119886119909

119903)minus1+119896119902119903

where 119896is a polynomial of degree at most 119896119902 minus 1

Proof In case (a) for we have1198910= 0 and119891

1= 119909119902 (41) take the

form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119892

119896(119909) = 0 (48)

whose solution is given by

119891119896(119909) = 119862

119896119909119896119902

minus 119909119896119902

int

119909

119904minus1minus119896119902

119892119896(119904) 119889119904 (49)

and taking into account that 119892119896(119909) that depends on 119891

119894for 119894 =

1 2 119896 minus 1 is at most of degree 119896119902 minus 1 we obtain that 119891119896is

a polynomial Moreover in this case system (5) becomes

= 119901119909 + 1199101206011(119909 119910)

= minus119902119910 + 1199101206012(119909 119910)

(50)

where 1206011and 120601

2are analytic functions

In case (b) we have that 1198910= 0 119891

1= 119909119902(1 + 119886119909

119903)minus119902119903 and

(41) takes the form

minus119896119902119891119896(119909) + 119909119891

1015840

119896(119909) + 119886119909

119903+11198911015840

119896(119909) + 119892

119896(119909) = 0 (51)

whose solution is given by

119891119896(119909) = 119909

119896119902(1 + 119886119909

119903)minus119896119902119903

times (119862119896minus int

119909

119904minus1minus119896119902

(1 + 119886119904)minus1+119896119902119903

119892119896(119904) 119889119904)

(52)

and taking into account the form of 119892119896(119909) we obtain that

119891119896(119909) is also of the form 119891

119896(119909) = 119901

119896(119909)(1 + 119886119909

119903)minus119896119902119904 where

119901119896(119909) is a polynomial because the integral in (52) does not

give logarithmic terms

Notice that in fact statement (a) is contained in statement(b) However we give the two statements in a separateform in order to be applied directly to different examplesCorollary 14 generalizes a lot of cases obtained in the liter-ature when some concrete families of polynomials systemswere studiedWe have notmade an exhaustive study of all thecases included in these large families but for instance containsthe cases (3) and (4) of Theorem 3 in [17] and case (3) ofTheorem 2 in [16]

In resume when we study a family that satisfies thenecessary conditions we must impose if the family systemhas a formal first integral of the form 119867

1or 1198672 If none of

them work we can look for the proper coordinates (1199111 1199112)

defined by the stable and instable manifold of the 119901 minus119902

resonant singular point In this way we must study if at leastone of the separatrices of the resonant point is algebraic Thebest case is when both are algebraicTherefore the problem tofind proper coordinates (119911

1 1199112) becomes a problem of finding

the invariant algebraic curves of system (5) passing throughthe 119901 minus119902 resonant singular point The existence of invariantalgebraic curves takes a leading role in the formal integrabilitytheory and is the base to find the proper coordinates systemThis is shown in the examples provided in [12] that aresolved by ad hoc methods and where the changes of variablesproposed to solve the examples are always invariant algebraiccurves of the original system passing through the originIn fact in [12] it recalled the Abhyankar-Moch theorem[37] This theorem establishes assuming the existence of aninvariant rational algebraic curve passing through the originthe existence of a rational invertible change of variables suchthat the invariant curve becomes one of the invariant axesIn this paper this result is generalized in the sense that any119901 minus119902 resonant singular point is formal orbitally equivalentto a system with 119910 = 0 as invariant algebraic curve seeLemma 5

6 Examples

In this section we present some examples where the resultsdeveloped in this paper are applied

Example 15 We first consider the 1 minus1 resonant quadraticsystem given by

= 119909 + 119886201199092+ 11988611119909119910 + 119886

021199102

= minus119910 + 119887201199092+ 11988711119909119910 + 119887

021199102

(53)

The aim is not to do an exhaustive study of when system (53)has a formal first integral in a neighborhood of the originIn fact the complex center cases were studied by Dulac see[34] The aim is only to show how to use the results givenin this paper If we apply directly Corollary 14 statement (a)we obtain 119887

20= 0 (this condition implies that 119910 = 0 be an

algebraic invariant curve of system (53)) and 11988620

= 11988711

= 0 Inthis case the quadratic system (53) reads

= 119909 + 11988611119909119910 + 119886

021199102

= minus119910 + 119887021199102

(54)

Abstract and Applied Analysis 9

and admits a formal first integral and by Theorem 2 system(54) has an analytic first integral in a neighborhood of theorigin

In fact the associated equation to system (54) is a linearequation with respect to 119909 and the system has a Darboux firstintegral of the form

119867(119909 119910)

= (11988702119910 minus 1)

minus1minus(1198861111988702)

times [211988602

minus 2 (1198860211988611

+ 1198860211988702) 119910

+ (1198863

11minus 119886111198872

02) 119909119910 + (119886

021198862

11+ 119886021198861111988702) 1199102]

(55)

Example 16 Consider the 1 minus1 resonant LotkandashVolterraplanar complex quartic system given by

= 119909 (1 minus 11988610119909 minus 11988601119910 minus 119886201199092minus 11988611119909119910

minus119886021199102minus 119886301199093minus 119886211199092119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988710119909 minus 11988701119910 minus 119887201199092minus 11988711119909119910

minus119887021199102minus 119887301199093minus 119887211199092119910 minus 119887121199091199102minus 119887031199103)

(56)

The LotkandashVolterra planar complex systems have been stud-ied in several papers see [7 11 13ndash15] In any case the LotkandashVolterra planar complex quartic system is an open problemcomputationally infeasible However with the results of thispaperwe are going to see that we can give some sufficient con-ditions to have an analytic first integral without computingany resonant focus value We propose a formal first integralof the form 119867

1 that is of the form (16) The recurrence that

gives119891119894(119909) is given by minus119894119891

119894(119909)+119909119891

1015840

119894(119909)+119892

119894(119909) = 0where 119892

119894(119909)

is(119894 minus 3) 119887

03119891119894minus3

(119909) + (119894 minus 2) 11988702119891119894minus2

(119909) + (119894 minus 2) 11988712119909119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) + (119894 minus 1) 11988711119909119891119894minus1

(119909)

+ (119894 minus 1) 119887211199092119891119894minus1

(119909)

+ 11989411988710119909119891119894(119909) + 119894119887

201199092119891119894(119909)

+ 119894119887301199093119891119894(119909) minus 119886

031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909) minus 119886

1111990921198911015840

119894minus1(119909)

minus 1198861011990921198911015840

119894(119909) minus 119886

2011990931198911015840

119894(119909) minus 119886

3011990941198911015840

119894(119909)

(57)

Now we impose that 119892119894(119909) must be of degree lei minus1 This

implies that 11988610

= 11988620

= 11988630

= 11988710

= 11988720

= 11988730

= 0 and11988721

= 11988621

= 11988711

= 11988611

= 0 and in this case the recurrence isminus119894119891119894(119909) + 119909119891

1015840

119894(119909) + 119892

119894(119909) = 0 where 119892

119894(119909) is given by

(119894 minus 3) 11988703119891119894minus3

(119909) + (119894 minus 2) (11988702

+ 11988712119909)119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) minus 119886031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909)

(58)

We obtain that its solution is given by

119891119894(119909) = 119862

119894119909119894minus 119909119894int

119909

119904minus1minus119894

119892119894(119904) 119889119904 (59)

and taking into account that 119892119894(119909) is of degree le 119894 minus 1 we can

establish the following result

Proposition 17 The system

= 119909 (1 minus 11988601119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988701119910 minus 119887021199102minus 119887121199091199102minus 119887031199103)

(60)

has an analytic first integral in a neighborhood of the origin

In fact Proposition 17 is a consequence of Proposition 11This technique can be applied to several families of polyno-mial systems and in fact gives a new mechanism to obtainresonant centers and also real centers for the complex systemsthat have a real preimage (when this preimage exists see eg[18])

Acknowledgments

A Algaba and C Garcıa are supported by aMICINNFEDERGrant no MTM2010-20907-C02-02 and by the Consejerıade Educacıon y Ciencia de la Junta de Andalucıa (projectsEXC2008FQM-872 TIC-130 and FQM-276) J Gine is par-tially supported by a MICINNFEDER Grant no MTM2011-22877 and by Generalitat de Catalunya Grant no 2009SGR381

References

[1] AAlgaba E Gamero andCGarcıa ldquoThe integrability problemfor a class of planar systemsrdquo Nonlinearity vol 22 no 2 pp395ndash420 2009

[2] A Algaba E Freire E Gamero and C Garcıa ldquoMonodromycenter-focus and integrability problems for quasi-homogeneouspolynomial systemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1726ndash1736 2010

[3] A Algaba N Fuentes and C Garcıa ldquoCenters of quasi-homogeneous polynomial planar systemsrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 419ndash431 2012

[4] A Algaba C Garcıa and M Reyes ldquoExistence of an inverseintegrating factor center problem and integrability of a class ofnilpotent systemsrdquo Chaos Solitons amp Fractals vol 45 no 6 pp869ndash878 2012

[5] J Chavarriga H Giacomini J Gine and J Llibre ldquoOn theintegrability of two-dimensional flowsrdquo Journal of DifferentialEquations vol 157 no 1 pp 163ndash182 1999

[6] H Zoładek ldquoThe problem of center for resonant singular pointsof polynomial vector fieldsrdquo Journal of Differential Equationsvol 137 no 1 pp 94ndash118 1997

[7] X Chen J Gine V G Romanovski and D S Shafer ldquoThe1 -q resonant center problem for certain cubic Lotka-Volterrasystemsrdquo Applied Mathematics and Computation vol 218 no23 pp 11620ndash11633 2012

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

and admits a formal first integral and by Theorem 2 system(54) has an analytic first integral in a neighborhood of theorigin

In fact the associated equation to system (54) is a linearequation with respect to 119909 and the system has a Darboux firstintegral of the form

119867(119909 119910)

= (11988702119910 minus 1)

minus1minus(1198861111988702)

times [211988602

minus 2 (1198860211988611

+ 1198860211988702) 119910

+ (1198863

11minus 119886111198872

02) 119909119910 + (119886

021198862

11+ 119886021198861111988702) 1199102]

(55)

Example 16 Consider the 1 minus1 resonant LotkandashVolterraplanar complex quartic system given by

= 119909 (1 minus 11988610119909 minus 11988601119910 minus 119886201199092minus 11988611119909119910

minus119886021199102minus 119886301199093minus 119886211199092119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988710119909 minus 11988701119910 minus 119887201199092minus 11988711119909119910

minus119887021199102minus 119887301199093minus 119887211199092119910 minus 119887121199091199102minus 119887031199103)

(56)

The LotkandashVolterra planar complex systems have been stud-ied in several papers see [7 11 13ndash15] In any case the LotkandashVolterra planar complex quartic system is an open problemcomputationally infeasible However with the results of thispaperwe are going to see that we can give some sufficient con-ditions to have an analytic first integral without computingany resonant focus value We propose a formal first integralof the form 119867

1 that is of the form (16) The recurrence that

gives119891119894(119909) is given by minus119894119891

119894(119909)+119909119891

1015840

119894(119909)+119892

119894(119909) = 0where 119892

119894(119909)

is(119894 minus 3) 119887

03119891119894minus3

(119909) + (119894 minus 2) 11988702119891119894minus2

(119909) + (119894 minus 2) 11988712119909119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) + (119894 minus 1) 11988711119909119891119894minus1

(119909)

+ (119894 minus 1) 119887211199092119891119894minus1

(119909)

+ 11989411988710119909119891119894(119909) + 119894119887

201199092119891119894(119909)

+ 119894119887301199093119891119894(119909) minus 119886

031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909) minus 119886

1111990921198911015840

119894minus1(119909)

minus 1198861011990921198911015840

119894(119909) minus 119886

2011990931198911015840

119894(119909) minus 119886

3011990941198911015840

119894(119909)

(57)

Now we impose that 119892119894(119909) must be of degree lei minus1 This

implies that 11988610

= 11988620

= 11988630

= 11988710

= 11988720

= 11988730

= 0 and11988721

= 11988621

= 11988711

= 11988611

= 0 and in this case the recurrence isminus119894119891119894(119909) + 119909119891

1015840

119894(119909) + 119892

119894(119909) = 0 where 119892

119894(119909) is given by

(119894 minus 3) 11988703119891119894minus3

(119909) + (119894 minus 2) (11988702

+ 11988712119909)119891119894minus2

(119909)

+ (119894 minus 1) 11988701119891119894minus1

(119909) minus 119886031199091198911015840

119894minus3(119909) minus 119886

021199091198911015840

119894minus2(119909)

minus 1198861211990921198911015840

119894minus2(119909) minus 119886

011199091198911015840

119894minus1(119909)

(58)

We obtain that its solution is given by

119891119894(119909) = 119862

119894119909119894minus 119909119894int

119909

119904minus1minus119894

119892119894(119904) 119889119904 (59)

and taking into account that 119892119894(119909) is of degree le 119894 minus 1 we can

establish the following result

Proposition 17 The system

= 119909 (1 minus 11988601119910 minus 119886121199091199102minus 119886031199103)

= minus119910 (1 minus 11988701119910 minus 119887021199102minus 119887121199091199102minus 119887031199103)

(60)

has an analytic first integral in a neighborhood of the origin

In fact Proposition 17 is a consequence of Proposition 11This technique can be applied to several families of polyno-mial systems and in fact gives a new mechanism to obtainresonant centers and also real centers for the complex systemsthat have a real preimage (when this preimage exists see eg[18])

Acknowledgments

A Algaba and C Garcıa are supported by aMICINNFEDERGrant no MTM2010-20907-C02-02 and by the Consejerıade Educacıon y Ciencia de la Junta de Andalucıa (projectsEXC2008FQM-872 TIC-130 and FQM-276) J Gine is par-tially supported by a MICINNFEDER Grant no MTM2011-22877 and by Generalitat de Catalunya Grant no 2009SGR381

References

[1] AAlgaba E Gamero andCGarcıa ldquoThe integrability problemfor a class of planar systemsrdquo Nonlinearity vol 22 no 2 pp395ndash420 2009

[2] A Algaba E Freire E Gamero and C Garcıa ldquoMonodromycenter-focus and integrability problems for quasi-homogeneouspolynomial systemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1726ndash1736 2010

[3] A Algaba N Fuentes and C Garcıa ldquoCenters of quasi-homogeneous polynomial planar systemsrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 419ndash431 2012

[4] A Algaba C Garcıa and M Reyes ldquoExistence of an inverseintegrating factor center problem and integrability of a class ofnilpotent systemsrdquo Chaos Solitons amp Fractals vol 45 no 6 pp869ndash878 2012

[5] J Chavarriga H Giacomini J Gine and J Llibre ldquoOn theintegrability of two-dimensional flowsrdquo Journal of DifferentialEquations vol 157 no 1 pp 163ndash182 1999

[6] H Zoładek ldquoThe problem of center for resonant singular pointsof polynomial vector fieldsrdquo Journal of Differential Equationsvol 137 no 1 pp 94ndash118 1997

[7] X Chen J Gine V G Romanovski and D S Shafer ldquoThe1 -q resonant center problem for certain cubic Lotka-Volterrasystemsrdquo Applied Mathematics and Computation vol 218 no23 pp 11620ndash11633 2012

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[8] C Christopher P Mardesic and C Rousseau ldquoNormalizableintegrable and linearizable saddle points for complex quadraticsystems in C2rdquo Journal of Dynamical and Control Systems vol9 no 3 pp 311ndash363 2003

[9] C Christopher and C Rousseau ldquoNormalizable integrableand linearizable saddle points in the Lotka-Volterra systemrdquoQualitative Theory of Dynamical Systems vol 5 no 1 pp 11ndash612004

[10] B Fercec X Chen and V G Romanovski ldquoIntegrabilityconditions for complex systems with homogeneous quinticnonlinearitiesrdquo The Journal of Applied Analysis and Computa-tion vol 1 no 1 pp 9ndash20 2011

[11] B Fercec J Gine Y Liu and V G Romanovski ldquoIntegrabilityconditions for Lotka-Volterra planar complex quartic systemshaving homogeneous nonlinearitiesrdquo Acta Applicandae Mathe-maticae vol 124 no 1 pp 107ndash122 2013

[12] A Fronville A P Sadovski and H Zołpolhk adek ldquoSolutionof the 1 minus2 resonant center problem in the quadratic caserdquoFundamenta Mathematicae vol 157 no 2-3 pp 191ndash207 1998

[13] J Gine Z Kadyrsizova Y Liu andVG Romanovski ldquoLineariz-ability conditions for Lotka-Volterra planar complex quarticsystems having homogeneous nonlinearitiesrdquo Computers ampMathematics with Applications vol 61 no 4 pp 1190ndash1201 2011

[14] J Gine and V G Romanovski ldquoLinearizability conditionsfor Lotka-Volterra planar complex cubic systemsrdquo Journal ofPhysics vol 42 no 22 Article ID 225206 2009

[15] J Gine and V G Romanovski ldquoIntegrability conditions forLotka-Volterra planar complex quintic systemsrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2100ndash21052010

[16] Z Hu V G Romanovski and D S Shafer ldquo1 minus3 resonant cen-ters on C2 with homogeneous cubic nonlinearitiesrdquo Computersamp Mathematics with Applications vol 56 no 8 pp 1927ndash19402008

[17] V G Romanovski X Chen and ZHu ldquoLinearizability of linearsystems perturbed by fifth degree homogeneous polynomialsrdquoJournal of Physics vol 40 no 22 pp 5905ndash5919 2007

[18] V G Romanovski and D S Shafer The Center and Cyclic-ity Problems A Computational Algebra Approach BirkhauserBoston Mass USA 2009

[19] A Algaba C Garcıa and M Reyes ldquoLocal bifurcation oflimit cycles and integrability of a class of nilpotent systems ofdifferential equationsrdquo Applied Mathematics and Computationvol 215 no 1 pp 314ndash323 2009

[20] A Algaba C Garcıa and M Reyes ldquoRational integrability oftwo-dimensional quasi-homogeneous polynomial differentialsystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 73 no 5 pp 1318ndash1327 2010

[21] A Algaba C Garcıa and M Reyes ldquoIntegrability of twodimensional quasi-homogeneous polynomial differential sys-temsrdquo The Rocky Mountain Journal of Mathematics vol 41 no1 pp 1ndash22 2011

[22] A Algaba C Garcia and M Reyes ldquoA note on analyticintegrability of planar vector fieldsrdquo European Journal of AppliedMathematics vol 23 pp 555ndash562 2012

[23] J Gine and X Santallusia ldquoEssential variables in the integrabil-ity problem of planar vector fieldsrdquo Physics Letters A vol 375no 3 pp 291ndash297 2011

[24] S Gravel and P Thibault ldquoIntegrability and linearizability ofthe Lotka-Volterra system with a saddle point with rationalhyperbolicity ratiordquo Journal of Differential Equations vol 184no 1 pp 20ndash47 2002

[25] C Liu G Chen and G Chen ldquoIntegrability of Lotka-Volterratype systems of degree 4rdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 1107ndash1116 2012

[26] C Liu G Chen and C Li ldquoIntegrability and linearizability ofthe Lotka-Volterra systemsrdquo Journal of Differential Equationsvol 198 no 2 pp 301ndash320 2004

[27] J-F Mattei and R Moussu ldquoHolonomie et integralespremieresrdquo Annales Scientifiques de lrsquoEcole Normale Superieurevol 13 no 4 pp 469ndash523 1980

[28] J Chavarriga H Giacomin J Gine and J Llibre ldquoLocalanalytic integrability for nilpotent centersrdquo Ergodic Theory andDynamical Systems vol 23 no 2 pp 417ndash428 2003

[29] J Gine ldquoOn the number of algebraically independent Poincare-Liapunov constantsrdquo Applied Mathematics and Computationvol 188 no 2 pp 1870ndash1877 2007

[30] J Gine ldquoHigher order limit cycle bifurcations from non-degenerate centersrdquoAppliedMathematics andComputation vol218 no 17 pp 8853ndash8860 2012

[31] J Gine and J Mallol ldquoMinimum number of ideal generatorsfor a linear center perturbed by homogeneous polynomialsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no12 pp e132ndashe137 2009

[32] M A Liapunov Probleme General de la Stabilite duMouvementvol 17 of Annals of Mathematics Studies Pricenton UniversityPress 1947

[33] H Poincare ldquoMemoire sur les courbes definies par les equationsdifferentiellesrdquo Journal de Mathematiques vol 37 pp 375ndash4221881 vol 8 pp 251ndash296 1882 Oeuvres de Henri Poincare volI Gauthier-Villars Paris pp 3ndash84 1951

[34] H Dulac ldquoDetermination et integration drsquoune certaine classedrsquoequations differentielles ayant pour point singulier un centrerdquoBulletin des Sciences Mathematiques vol 32 no 2 pp 230ndash2521908

[35] A D Bruno Local Methods in Nonlinear Differential EquationsSpringer Berlin Germany 1989

[36] A D Bruno Power Geometry in Algebraic and DifferentialEquations vol 57 of North-Holland Mathematical LibraryNorth-Holland Publishing Co Amsterdam The Netherlands2000

[37] S S Abhyankar and T T Moh ldquoEmbeddings of the line in theplanerdquo Journal fur die Reine und Angewandte Mathematik vol276 pp 148ndash166 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of