Research Article Twist Periodic Solutions in the ...downloads.hindawi.com/journals/aaa/2016/6084082.pdfTwist Periodic Solutions in the Relativistic Driven Harmonic Oscillator ... .We

Research ArticleTwist Periodic Solutions in the Relativistic DrivenHarmonic Oscillator

Daniel Nuacutentildeez and Andreacutes Rivera

Departamento de Ciencias Naturales y Matematicas Pontificia Universidad Javeriana Seccional Cali 26239 Cali Colombia

Correspondence should be addressed to Andres Rivera amriverajaverianacalieduco

Received 27 March 2016 Accepted 16 May 2016

Academic Editor Svatoslav Stanek

Copyright copy 2016 D Nunez and A Rivera This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We study the one-dimensional forced harmonic oscillator with relativistic effects Under some conditions of the parameters theexistence of a unique stable periodic solution is proved which is of twist typeThe results depend on a TwistTheorem for nonlinearHillrsquos equations which is established and proved here

1 Introduction and Main Results

In this paper we study the existence of a stable periodicsolution (periodic response) in the one-dimensional forcedharmonic oscillator with relativistic effects

( 1198981199091015840radic1 minus 119909101584021198882)

1015840

+ 119896119909 = minus1198650cos120596119905 (1)

where 119898 gt 0 is the mass at rest 119888 gt 0 is the speed oflight in the vacuum 119896 gt 0 is the spring stiffness coefficientand 119865

0 120596 are the amplitude and frequency of the external

force Physically we are assuming a basic principle of specialrelativity the mass of a moving object is not constant butdepends on its velocityThis equation can be derived from anappropriate Lagrangian or Hamiltonian formulation [1] TheHamiltonian association of this system is

119867(119909 119910) = radic11991021198882 + 11989821198884 + 121198961199092 + 1199091198650 cos120596119905 (2)

The existence of chaotic behavior in the relativisticharmonic oscillator has been investigated numerically in[2] More generally the existence of chaotic dynamics ofrelativistic particles (relativistic chaos) has been reported inmany different contexts [3ndash6]

From amore mathematical perspective (1) can be seen asa singular 120601-Laplacian oscillator [7] then Landesman-Lazercondition holds and there exists a periodic solution for all the

values of the parameters In other words relativistic effectskill the classical linear resonance phenomenon

Chu et al have proven in [8] the stability of the equilib-rium 119909 equiv 0 of the relativistic pendulum with variable lengthWe use a similar approach in order to prove the stabilityof a periodic solution for (1) obtained via lower and uppersolutions in the reversed order (see Section 2)

On the other hand in recent years some results aboutsimilar oscillators with relativistic effects like the relativisticforced pendulum [9ndash11] have been published There theauthors have proven the existence andmultiplicity of periodicsolutions in the relativistic forced pendulum by variationaland topological methods

Our aim is to study the existence and stability of periodicsolutions for the relativistic harmonic oscillator (1) withperiod 119879 = 2120587120596 We search 119879-periodic solutions so-calledof twist type meaning that its Floquet multipliers are not realand are not 119899th-root of the unity for 119899 = 1 2 3 4 and itsfirst Birkhoff rsquos coefficient is different to zero (see Section 4for definitions)

It is a well known fact that the appearance of 119879-periodicsolution of twist type typically exhibits a KAM scenarioaround it that is existence of 119899119879-periodic solutions for119899 being arbitrary large in all neighborhood of it some ofwhich will be elliptic and the others will be hyperbolic Thehyperbolic solutions have generically transversal intersec-tions between their associated stable and unstable manifolds[12 13]

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2016 Article ID 6084082 7 pageshttpdxdoiorg10115520166084082

2 Abstract and Applied Analysis

The genericity is understood relative to certain topologyconstructed via jets of functions [13] So close enough tothose intersections certain invariant compact sets for thePoincare transformation almost always (generically speak-ing) appear called Smalersquos Horseshoe A popular result in thetheory of dynamical systems is that the dynamic on SmalersquosHorseshoe presents dense periodic orbits and sensitivity withrespect to initial conditions and random itineraries (chaoticbehavior)

According to this the proof of the existence of periodicsolutions of twist type is a first step in the comprehension ofthe chaotic behavior numerically evidenced in [2]

Theorem 1 is the main result of this paper

Theorem 1 Assume that the parameters119898 119896120596 and1198650satisfy

the following conditions

(H1) 119896119898 lt 120596216(H2) 119865

0lt (14)119898119888120596

(H3) ((119898119888120596)19[(119898119888120596)2 minus 1611986520]1201205871198652

0(119888211989821205962 +

411986520)192) sin(6120587120596121198981198883211989612(119888211989821205962 + 4119865

0

2)34) gt1and then the driven relativistic harmonic oscillator (1) hasunique 2120587120596-periodic solution which is of twist type andtherefore Lyapunov stable

Hypothesis (H3) may look rather weird however it givessome interesting corollaries in a direct way

Corollary 2 With fixed120596 1198961198650 and 119888 in (1) there exists119872

0equiv119872

0(120596 119896 119865

0 119888) gt 0 such that if 119898 gt 119872

0then the conclusion of

Theorem 1 holds

Corollary 3 With fixed 119898 119896 1198650 and 119888 in (1) there exists a

critical frequency 1205960equiv 1205960(119898 119896 119865

0 119888) such that if 120596 gt 120596

0then

the conclusion of Theorem 1 holds

Both corollaries follow easily by passing to the limit in theconditions ofTheorem 1Moreover the critical values119872

01199080

can be numerically computed For instance 1198720(1 1 1 1) ≃4165497 and 120596

0(1 1 1 1) ≃ 2435658

The rest of the paper is organized as follows Section 2 isdevoted to analyze the existence of a periodic solution As it iscommented before such existence is direct from the results of[7] althoughwe present an alternativemethodwhich gives ussome bounds for the solutionThe key idea is the reduction of(1) to an equivalent Newtonian nonlinear oscillatorThen theuse of upper and lower solutions provides the existence aswellas some bounds which are necessary in the sequel Section 3analyzes the uniqueness and linear stability of the periodicsolution Finally in Section 4 a new Twist Theorem in theline of those presented in [14ndash16] is proved which is appliedin Section 5 in order to prove Theorem 1 Notice that it isnot possible to apply directlyTheorem 32 in [16] because theestimate (339) is not correctThe right estimate is establishedandproved in Lemma 8 and thiswill be fundamental in orderto establish the twist criteria (Theorem 7)

2 Existence of Periodic Solutions

As we mentioned in Introduction and Main Results theexistence of a periodic solution for (1) is a direct consequenceof the results contained in [7] We use here an alternativeapproach based on upper and lower solutions because itprovides explicit bounds for the solutionwhichwill be cruciallater

For the Hamiltonian (2) we get Hamiltonrsquos equations

1199091015840 = 1198882119910radic11991021198882 + 11989821198884

1199101015840 = minus119896119909 minus 1198650cos120596119905

(3)

By deriving in the second equation this system is equivalentto the second-order equation

11991010158401015840 + 119891 (119910) = 1205961198650sin120596119905 (4)

with

119891 (119910) = 1198961198882119910radic11991021198882 + 11989821198884 (5)

More precisely notice that the associated first-order systemfor (4) with state variables 119906 = 119910 and V = 1199101015840 can be obtainedfrom (3) by means of the following symplectic changes ofvariables with multiplier

119906 = 119910V = minus119896119909 minus 119865

0cos (120596119905) (6)

Thus it is clear that the study of stability in both systems isequivalent because the Poincare mappings are conjugatedTherefore the dynamics of a driven relativistic harmonicoscillator are equivalent to those of a driven nonrelativisticoscillator with the potential 119881(119910) = 119896radic11991021198882 + 11989821198884 ThisNewtonian equation has minimal period 119879 = 2120587120596 andwe are interested in some key dynamical aspects like theexistence of periodic solutions and its stability propertiesThefollowing result ensures us that (4) (as well as therefore (1))has at least a 119879-periodic solutionProposition 4 Let one assume that 4119896 lt 1198981205962 Then (4) hasa 119879-periodic solution 120593 such that

minus1198650120596 (1 + sin120596119905) lt 120593 (119905) lt 1198650120596 (1 minus sin120596119905) (7)

for all 119905Proof It follows from the classical theory of upper and lowersolutions [17 Theorem 41] It is easy to verify that 120572(119905) =(1198650120596)(1 minus sin120596119905) is a lower solution and 120573(119905) = minus(119865

0120596)(1 +

sin120596119905) is an upper solution such that 120573(119905) lt 120572(119905) for all 119905Moreover note that |1198911015840(119904)| le 119896119898 lt (120587119879)2 which is thecondition for existence of a solution between the upper andthe lower solution on the reversed order

Abstract and Applied Analysis 3

Of course the periodic solution 120593 of (4) provides aperiodic solution to the original equation (1) given by 119909(119905) =minus(1205931015840(119905) + 119865

0cos120596119905)119896

3 Uniqueness and Linear Stability

In this section we study the stability of the periodic solutionfound in the previous section in the linear sense Let usfix a 119879-periodic solution 120593 of (4) which always exists byProposition 4 and we translate it to the origin making thecanonical changes of variables 119909 = 119910 minus 120593(119905)Thus we lead tothe equivalent equation

11990910158401015840 + 119891 (119909 + 120593 (119905)) minus 119891 (120593 (119905)) = 0 (8)

Now the equilibrium 119909 equiv 0 is a solution The linearization of119909 = 0 is Hillrsquos equation11990910158401015840 + 119886 (119905) 119909 = 0 (9)

where

119886 (119905) = 11989611988831198982(1205932 (119905) + 11989821198882)32 (10)

By definition 120593 is said to be elliptic if the Floquet multipliersofHillrsquos equation (9) are complex conjugate numbers differentfrom plusmn1 An elliptic solution is in particular linearly stable

The following bounds over 119886(119905) are easily deduced12059021le 119886 (119905) le 1205902

2 (11)

with

12059021= 119896119898 (1 + ( 2119865

0119888119898120596)2)minus32

12059022= 119896119898

(12)

Now we can formulate and prove the following result

Proposition 5 Assume that

4119896 lt 1198981205962 (13)

Then (4) has a unique 119879-periodic solution 120593 which is elliptic

Proof of Proposition 5 From (13) and (11)-(12) we deduce that

0 lt 119886 (119905) lt (120587119879)2 forall119905 isin R (14)

Then one application of the classical Lyapunov-Zukovskiistability criterion (see [18 19]) implies the ellipticity of (9)

For the uniqueness suppose that 120595(119905) is another periodicsolution Then 120601 minus 120595 is a 119879-periodic solution to Hillrsquosequation

11990910158401015840 + ℎ (119905) 119909 = 0 (15)

where

ℎ (119905) =

119891 (120593 (119905)) minus 119891 (120595 (119905))120593 (119905) minus 120595 (119905) if 120593 (119905) = 120595 (119905)

1198911015840 (120593 (119905)) otherwise(16)

By applying the mean value theorem and taking into accountthat 0 lt 1198911015840 lt 119896119898 we get

0 lt ℎ (119905) lt 119896119898 lt (120587119879)

2

(17)

for all 119905 Again it follows that (15) is elliptic therefore the only119879-periodic solution is the trivial one so 120593 = 120595 This provesthe uniqueness

4 Nonlinear Lyapunovrsquos Stability

The system under study is conservative so the stability inthe sense of Lyapunov can not be directly derived fromthe first approximation because of the possible synchronizedinfluence of higher terms leading to resonance After theworks of Siegel and Moser [12] it is well known that thestability in the nonlinear sense depends generically on thethird approximation of the periodic solution So we willfocus on the third approximation for the reduced problem (4)around the periodic solution 120593

From the point of view of KAM theory ([12 20 21])the nonlinear terms of Taylorrsquos expansion around a givenperiodic solution are taken into account to decide the kindof dynamic rising around such a solution The basic ideaconsists in expressing the system in suitable geometricalcoordinates as a perturbation of a canonical system whichis integrable and therefore possesses invariant tori near tothe periodic solution These invariant tori are persistentunder perturbations and produce jails or barriers for theflux trapping the orbits inside As a by-product it obtainedthe typical KAM scenario around the periodic solution (see[12 22]) The effective existence of a homoclinic transversalpoint for the Poincare mapping P (which generates SmalersquosHorseshoe dynamics) is an interesting open question for thismodel However it is known that this property is generic forarea preserving mappings [13 22]

More recently these ideas have taken a renewal intereststarting from some Ortegarsquos works [14 23 24] that provide uswith some stability criteria based on the third approximationSome related references are [15 16 25ndash29] We follow thisapproach and give a new stability criterion in line with thosepresented in [15 Theorem 22] and [16 Theorems 31 and32]

Notice that the estimate (339) in [16] and as a conse-quence the condition (349) in Theorem 32 of that papermust be carefully reviewed The right estimate is statedand proved in this section (Lemma 8) All these facts havemotivated the rewriting of a clean criterion that works at leastin the so-called first stability zone

We consider the nonlinear Hill equation

11990910158401015840 + 119886 (119905) 119909 + 119887 (119905) 1199092 + 119888 (119905) 1199093 + 119877 (119905 119909) = 0 (18)


where the functions 119886 119887 119888 R119879Z rarr R are continuous 119887and 119888 are not both identically zero and the remainder 119877 isin1198620infin(R119879Z) times (minus120598 120598) 120598 gt 0 satisfies

120597119896119877120597119909119896 (119905 0) = 0 119896 = 0 1 2 3 forall119905 isin R (19)

The solution 119909 equiv 0 is an equilibrium of (18)The linearization of (18) at 119909 = 0 is Hillrsquos equation

11990910158401015840 + 119886 (119905) 119909 = 0 (20)

Let119872 be the monodromymatrix of (20)The eigenvalues12058212

of119872 are called theFloquetmultipliers of (20)TheFloquetmultipliers of (20) satisfy

12058211205822= 1 (21)

In a classical terminology it is said that (20) (or 119909 = 0) iselliptic if 120582

2= 1205821isin C1 plusmn1 parabolic if 120582

12= plusmn1 and

hyperbolic if |12058212| = 1 respectively In the hyperbolic case

not only is the linear equation unstable but also 119909 = 0 likesolution to (18)

Given 119899 isin N we say that the equilibrium 119909 = 0 of (18)is 119899-resonant if it is elliptic and the Floquet multipliers satisfy120582119899119894= 1We say that119909 = 0 is strongly resonant if it is 119899-resonant

for 119899 = 3 or 4The Poincaremapping associatedwith (18) is defined near

the origin by

P (119909 119910) = 120593 (119879 119909 119910) (22)

where 120593(119905 119909 119910) is the unique solution to (18) such that

120593 (0 119909 119910) = 1199091205931015840 (0 119909 119910) = 119910 (23)

Note that P(0 0) = (0 0) and then the stability of119909 equiv 0 (like the 119879-periodic solution to (18)) is equivalentto the stability of (0 0) as fixed point of P The otherelementary property of the Poincare map states thatP1015840(0 0)is a monodromy matrix for (20) and then its eigenvaluesare the Floquet multipliers of (20) If (20) is elliptic and notstrongly resonant by Birkhoff Normal Form Theorem thereexists a canonical change of variables 119911 = Φ(120585) and 119911 = (119909 119910)such that P adopts in the new coordinates the followingform

Plowast (120585) = (Φminus1 ∘P ∘ Φ) (120585) = 119877 [120579 + 120573 100381610038161003816100381612058510038161003816100381610038162] (120585) + 1198744 (24)

where 119877[120575] denotes the rotation of angle 120575 120582 = 119890plusmn119894120579 are theFloquet multipliers and 119874

4indicates a term that is 119874(|120585|4)

when 120585 rarr 0 The coefficient 120573 is called the first twistcoefficient and plays a central role in the stability From theTwistTheorem it follows that if 120573 = 0 then (0 0) is stable (see[12 chapter 3])

Definition 6 We say that the equilibrium 119909 equiv 0 of (18) is oftwist type if it is elliptic and not strongly resonant and theassociated first twist coefficient 120573 = 0

Notice that according to this definition all equilibriumof twist type is Lyapunov stable Also it is known from thegeneral theory that an equilibriumof twist type exhibitsKAMdynamics around it as was mentioned in Introduction andMain Results

The twist coefficient 120573 has an explicit formula and it isproportional to the integral quantity (see [14 16])

120573lowast = minus38 int119879

0

119903 (119905)4 119888 (119905) 119889119905 +∬[0119879]2

119887 (119905) 119887 (119904) 1199033 (119905)sdot 1199033 (119904) 120594

2(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904

(25)

where 119903(119905) is the unique positive 119879-periodic solution to theEmarkov-Pinney equation ([16 Lemma 33])

11991010158401015840 + 119886 (119905) 119910 = 11199103 (26)

the function120601 is any primitive of 11199032 and the kernel function1205942is defined by

1205942(119909) = 2cos3 (119909 minus 1205792) + 3 cos 120579 cos (119909 minus 1205792)

8 sin (31205792) 119909 isin ]0 120579]

(27)

The main result of this section is as follows

Theorem 7 Assume that for (18) the third coefficient 119888(119905) lt 0forall119905 isin R Choose 119888lowastgt 0 and 119887lowast gt 0 such that

(i) minus119888(119905) gt 119888lowastgt 0 forall119905 isin R

(ii) |119887(119905)| le 119887lowast forall119905 isin R

Assume that the following conditions hold

0 lt 12059021le 119886 (119905) le 120590

2

2 lt ( 1205872119879)2 forall119905 isin R (28)

119888lowastgt 10119879120590

2

7

3 sin ((31198792) 1205901) 12059081

119887lowast2 (29)

Then the equilibrium 119909 equiv 0 of (18) is of twist typeLemma 8 Assume that the condition (28) ofTheorem 7 holdsThen

1205901211205902

le 119903 (119905) le 1205901221205901

forall119905 isin R (30)

Proof The function 119903(119905) can be obtained by the followingrelation ([16 Section 32])

119903 (119905) = 120572119903lowast (119905 minus 11990501205722 ) (31)

for certain time scaling 120591 = (119905minus1199050)1205722 which transformsHillrsquos

equation 11990910158401015840 + 119886(119905) = 0 into another one11990910158401015840 + 119886lowast (120591) 119909 = 0 119886lowast (120591) = 1205724119886 (119905

0+ 1205722120591) (32)


with new period 119879lowast = 1198791205722 such that it is 119877-elliptic thatis the associated monodromy matrix is a rigid rotation (see[14 Propositon 7]) The function 119903lowast is defined as 119903lowast = |Ψ|whereΨ is the complex solution to (32)with initial conditionsΨ(0) = 1 and Ψ1015840(0) = 119894 Clearly 119886lowast satisfies the samecondition (28) of Theorem 7 with new constants 120590lowast

119894= 1205722120590

119894119894 = 1 2 So 119886lowast holds the conditions of Lemmas 42 and 43(ii)

of [15] Combining these lemmas we obtain

112057221205902

le 119903lowast le 112057221205901

(33)

So finally from the relation (31) we obtain the followinguniform bounds on 119903

11205721205902

le 119903 le 11205721205901

(34)

From Lemma 43(ii) we know that 120590minus122

le 120572 le 120590minus121

sofinally we arrive from (34) to the required inequality

Proof of Theorem 7 The proof ofTheorem 7 follows basicallythe lines of [16] For a function 119891(119905) let 119891+ = max119891 0 and119891minus = maxminus119891 0 denote the positive and negative part of 119891So we can write 119891 = 119891+ minus 119891minus

Condition (28) implies (see [15]) that 119909 equiv 0 is elliptic andnot strongly resonant therefore the first twist coefficient 120573 iswell defined Remember that 120573 is proportional to 120573lowast given by(25) In order to proveTheorem 7 it is sufficient to show that120573lowast gt 0

On the other hand from (28) we also deduce thatthe rotation number 120588 associated with Hillrsquos equation (20)satisfies

1205901le 120588 le 120590

2lt 1205872119879 (35)

The characteristic exponent is by definition 120579 = 119879120588 Thus

1205901119879 le 120579 le 120590

2119879 lt 120587

2 (36)

Let 1199031= 1205901211205902and 1199032= 1205901221205901denote respectively the

lower and upper bound of 119903(119905) given by Lemma 8 So from(25) one deduces that

120573lowast ge 3811987911990341119888lowast +∬[0119879]2 (119887

+ (119905) minus 119887minus (119905))sdot (119887+ (119904) minus 119887minus (119904)) 1199033 (119905) 1199033 (119904)sdot 1205942(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904 ge 3

811987911990341119888lowastminus∬[0119879]2

119887+ (119905) 119887minus (119904) 1199033 (119905) 1199033 (119904)sdot 1205942(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904 minus int

[0119879]2

119887minus (119905) 119887+ (119904)sdot 1199033 (119905) 1199033 (119904) 120594

2(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904

(37)

because the function 1205942is positive in (0 120579] sub ]0 120587[ (see (27)

and (36)) On the other hand an upper bound for 1205942was

computed in [16]

10038161003816100381610038161205942 (119909)1003816100381610038161003816 le 2 + 3 cos 1205798 sin(31205792) (119909 isin ]0 120579] 120579 isin ]0 1205872 [) (38)

Thus using (36) and from themonotonocity of sin and cos in]0 1205872[ we get10038161003816100381610038161205942 (119909)1003816100381610038161003816 le 2 + 3 cos119879120590

18 sin ((31198792) 1205901) (39)

Going back to 120573lowast now we can deduce that

120573lowast ge 3811987911990341119888lowast minus 2119879211990326119887lowast

2 58 sin ((31198792) 120590

1) gt 0 (40)

where the last inequality is a consequence of condition (29)

5 Applications to the Relativistic Oscillator

In Section 2 we prove the existence of a unique 2120587120596-periodic elliptic solution 120593 for the problem (4) under asuitable hypothesis over the parameters 119896 and 119898 moreconcretely if 4119896 lt 1198981205962 One more time we emphasize thatthis problem is equivalent to the driven relativistic harmonicoscillator (1)

The third approximation for (4) around the periodicsolution 120593 is given by

11990910158401015840 + 119886 (119905) 119909 + 119887 (119905) 1199092 + c (119905) 1199093 + sdot sdot sdot = 0 (41)

where

119886 (119905) = 11989611988831198982(1205932 (119905) + 11989821198882)32

119887 (119905) = minus311989611989821198883120593 (119905)2 (1205932 (119905) + 11989821198882)52

119888 (119905) = 11989611989821198883 (41205932 (119905) minus 11989821198882)2 (1205932 (119905) + 11989821198882)72

(42)

Proof of Theorem 1 From Proposition 5 and hyphothesis(H1) we have a unique 119879-periodic solution 120593which is ellipticand not strongly resonant and verifying the bound (7) Inparticular

1003816100381610038161003816120593 (119905)1003816100381610038161003816 lt 21198650120596 (43)

In order to prove that 120593 is of twist type we will applyTheorem 7 to the third approximation (41) By hyphothesis


(H2) we get 119888(119905) lt 0 forall119905 isin R Hence the constants involvedinTheorem 7 can be taken as

12059021= 119896119898 (1 + ( 2119865

0119888119898120596)2)minus32

12059022= 119896119898

119887lowast = 3119896119865012059611989831198882

119888lowast= 11989611989821198883 (11989821198882 minus 161198652

01205962)

2 (11989821198882 + 4119865201205962)72

(44)

After several tedious computations one can see that theinequalities (H3) of Theorem 1 and (29) of Theorem 7are equivalent The application of Theorem 7 finishes theproof

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Capital Semilla 2014-2015 project00004025 Pontificia Universidad Javeriana Seccional CaliCali Colombia

References

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[2] J-H Kim and H-W Lee ldquoRelativistic chaos in the drivenharmonic oscillatorrdquo Physical Review E vol 51 no 2 pp 1579ndash1581 1995

[3] M Billardon ldquoStorage ring free-electron laser and chaosrdquoPhysical Review Letters vol 65 no 6 pp 713ndash716 1990

[4] C Chen andRCDavidson ldquoChaotic particle dynamics in free-electron lasersrdquo Physical ReviewA vol 43 no 10 pp 5541ndash55541991

[5] A A Chernikov T Tel G Vattay and G M Zaslavsky ldquoChaosin the relativistic generalization of the standard maprdquo PhysicalReview A vol 40 no 7 pp 4072ndash4076 1989

[6] W P Leemans C Joshi W B Mori C E Clayton and T WJohnston ldquoNonlinear dynamics of driven relativistic electronplasma wavesrdquo Physical Review A vol 46 no 8 pp 5112ndash51221992

[7] C Bereanu and J Mawhin ldquoExistence and multiplicity resultsfor some nonlinear problems with singular 120601-Laplacianrdquo Jour-nal of Differential Equations vol 243 no 2 pp 536ndash557 2007

[8] J Chu J Lei and M Zhang ldquoThe stability of the equilibriumof a nonlinear planar system and application to the relativisticoscillatorrdquo Journal of Differential Equations vol 247 no 2 pp530ndash542 2009

[9] S Maro ldquoPeriodic solutions of a forced relativistic pendulumvia twist dynamicsrdquo Topological Methods in Nonlinear Analysisvol 42 no 1 pp 51ndash75 2013

[10] H Brezis and J Mawhin ldquoPeriodic solutions of the forcedrelativistic pendulumrdquo Differential and Integral Equations vol23 no 9-10 pp 801ndash810 2010

[11] C Bereanu and P J Torres ldquoExistence of at least two periodicsolutions of the forced relativistic pendulumrdquo Proceedings of theAmerican Mathematical Society vol 140 no 8 pp 2713ndash27192012

[12] C L Siegel and J K Moser Lectures on Celestial MechanicsSpringer New York NY USA 1971

[13] C Genecand ldquoTransversal homoclinic orbits near elliptic fixedpoints of area-preserving diffeomorphisms of the planerdquo inDynamics Reported Expositions in Dynamical Systems vol 2Springer Berlin Germany 1993

[14] ROrtega ldquoPeriodic solutions of aNewtonian equation stabilityby the third approximationrdquo Journal of Differential Equationsvol 128 no 2 pp 491ndash518 1996

[15] D Nunez ldquoThe method of lower and upper solutions and thestability of periodic oscillationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 51 no 7 pp 1207ndash1222 2002

[16] J Lei X Li P Yan and M Zhang ldquoTwist character of the leastamplitude periodic solution of the forced pendulumrdquo SIAMJournal on Mathematical Analysis vol 35 no 4 pp 844ndash8672003

[17] C De Coster and P Habets ldquoUpper and lower solutions in thetheory of ODE boundary value problems classical and recentresultsrdquo in Non Linear Analysis and Boundary Value Problemsfor Ordinary Differential Equations F Zanolin Ed vol 371 ofCISM-ICMS Courses and Lectures pp 1ndash78 Springer ViennaAustria 1996

[18] W Magnus and S Winkler Hillrsquos Equation Dover New YorkNY USA 1979

[19] V M Starzinskii ldquoA survey of works on the conditions ofstability of the trivial solution of a system of linear differentialequations with periodic coefficientsrdquo American MathematicalSociety Translations vol 1 pp 189ndash237 1955

[20] V ArnoldMethodes Mathematiques de la Mecanique ClassiqueMir Moscow Russia 1976

[21] J Moser On Invariant Curves of Area-Preserving Mappings ofan Annulus Nachrichten der Akademie der Wissenschaften inGottingen II Mathematisch-Physikalische Klasse II Vanden-hoeck amp Ruprecht Gottingen Germany 1962

[22] S E Newhouse ldquoQuasi-elliptic periodic points in conservativedynamical systemsrdquo American Journal of Mathematics vol 99no 5 pp 1061ndash1087 1977

[23] R Ortega ldquoThe stability of the equilibrium of a nonlinear Hillrsquosequationrdquo SIAM Journal on Mathematical Analysis vol 25 no5 pp 1393ndash1401 1994

[24] R Ortega ldquoThe twist coefficient of periodic solutions of atime-dependent Newtonrsquos equationrdquo Journal of Dynamics andDifferential Equations vol 4 no 4 pp 651ndash665 1992

[25] B Liu ldquoThe stability of the equilibrium of a conservativesystemrdquo Journal of Mathematical Analysis and Applications vol202 no 1 pp 133ndash149 1996

[26] D Nunez and R Ortega ldquoParabolic fixed points and stabilitycriteria for non-linear Hillrsquos equationrdquo Zeitschrift fur Ange-wandte Mathematik und Physik ZAMP vol 51 no 6 pp 890ndash911 2000

[27] D Nunez and P J Torres ldquoPeriodic solutions of twist type ofan earth satellite equationrdquoDiscrete and Continuous DynamicalSystems vol 7 no 2 pp 303ndash306 2001


[28] DNunez and P J Torres ldquoStable odd solutions of some periodicequations modeling satellite motionrdquo Journal of MathematicalAnalysis and Applications vol 279 no 2 pp 700ndash709 2003

[29] D Nunez and P J Torres ldquoKAM dynamics and stabilizationof a particle sliding over a periodically driven curverdquo AppliedMathematics Letters vol 20 no 6 pp 610ndash615 2007

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The genericity is understood relative to certain topologyconstructed via jets of functions [13] So close enough tothose intersections certain invariant compact sets for thePoincare transformation almost always (generically speak-ing) appear called Smalersquos Horseshoe A popular result in thetheory of dynamical systems is that the dynamic on SmalersquosHorseshoe presents dense periodic orbits and sensitivity withrespect to initial conditions and random itineraries (chaoticbehavior)

According to this the proof of the existence of periodicsolutions of twist type is a first step in the comprehension ofthe chaotic behavior numerically evidenced in [2]

Theorem 1 is the main result of this paper

Theorem 1 Assume that the parameters119898 119896120596 and1198650satisfy

the following conditions

(H1) 119896119898 lt 120596216(H2) 119865

0lt (14)119898119888120596

(H3) ((119898119888120596)19[(119898119888120596)2 minus 1611986520]1201205871198652

0(119888211989821205962 +

411986520)192) sin(6120587120596121198981198883211989612(119888211989821205962 + 4119865

0

2)34) gt1and then the driven relativistic harmonic oscillator (1) hasunique 2120587120596-periodic solution which is of twist type andtherefore Lyapunov stable

Hypothesis (H3) may look rather weird however it givessome interesting corollaries in a direct way

Corollary 2 With fixed120596 1198961198650 and 119888 in (1) there exists119872

0equiv119872

0(120596 119896 119865

0 119888) gt 0 such that if 119898 gt 119872

0then the conclusion of

Theorem 1 holds

Corollary 3 With fixed 119898 119896 1198650 and 119888 in (1) there exists a

critical frequency 1205960equiv 1205960(119898 119896 119865

0 119888) such that if 120596 gt 120596

0then

the conclusion of Theorem 1 holds

Both corollaries follow easily by passing to the limit in theconditions ofTheorem 1Moreover the critical values119872

01199080

can be numerically computed For instance 1198720(1 1 1 1) ≃4165497 and 120596

0(1 1 1 1) ≃ 2435658

The rest of the paper is organized as follows Section 2 isdevoted to analyze the existence of a periodic solution As it iscommented before such existence is direct from the results of[7] althoughwe present an alternativemethodwhich gives ussome bounds for the solutionThe key idea is the reduction of(1) to an equivalent Newtonian nonlinear oscillatorThen theuse of upper and lower solutions provides the existence aswellas some bounds which are necessary in the sequel Section 3analyzes the uniqueness and linear stability of the periodicsolution Finally in Section 4 a new Twist Theorem in theline of those presented in [14ndash16] is proved which is appliedin Section 5 in order to prove Theorem 1 Notice that it isnot possible to apply directlyTheorem 32 in [16] because theestimate (339) is not correctThe right estimate is establishedandproved in Lemma 8 and thiswill be fundamental in orderto establish the twist criteria (Theorem 7)

2 Existence of Periodic Solutions

As we mentioned in Introduction and Main Results theexistence of a periodic solution for (1) is a direct consequenceof the results contained in [7] We use here an alternativeapproach based on upper and lower solutions because itprovides explicit bounds for the solutionwhichwill be cruciallater

For the Hamiltonian (2) we get Hamiltonrsquos equations

1199091015840 = 1198882119910radic11991021198882 + 11989821198884

1199101015840 = minus119896119909 minus 1198650cos120596119905

(3)

By deriving in the second equation this system is equivalentto the second-order equation

11991010158401015840 + 119891 (119910) = 1205961198650sin120596119905 (4)

with

119891 (119910) = 1198961198882119910radic11991021198882 + 11989821198884 (5)

More precisely notice that the associated first-order systemfor (4) with state variables 119906 = 119910 and V = 1199101015840 can be obtainedfrom (3) by means of the following symplectic changes ofvariables with multiplier

119906 = 119910V = minus119896119909 minus 119865

0cos (120596119905) (6)

Thus it is clear that the study of stability in both systems isequivalent because the Poincare mappings are conjugatedTherefore the dynamics of a driven relativistic harmonicoscillator are equivalent to those of a driven nonrelativisticoscillator with the potential 119881(119910) = 119896radic11991021198882 + 11989821198884 ThisNewtonian equation has minimal period 119879 = 2120587120596 andwe are interested in some key dynamical aspects like theexistence of periodic solutions and its stability propertiesThefollowing result ensures us that (4) (as well as therefore (1))has at least a 119879-periodic solutionProposition 4 Let one assume that 4119896 lt 1198981205962 Then (4) hasa 119879-periodic solution 120593 such that

minus1198650120596 (1 + sin120596119905) lt 120593 (119905) lt 1198650120596 (1 minus sin120596119905) (7)

for all 119905Proof It follows from the classical theory of upper and lowersolutions [17 Theorem 41] It is easy to verify that 120572(119905) =(1198650120596)(1 minus sin120596119905) is a lower solution and 120573(119905) = minus(119865

0120596)(1 +

sin120596119905) is an upper solution such that 120573(119905) lt 120572(119905) for all 119905Moreover note that |1198911015840(119904)| le 119896119898 lt (120587119879)2 which is thecondition for existence of a solution between the upper andthe lower solution on the reversed order



0cos120596119905)119896



11990910158401015840 + 119891 (119909 + 120593 (119905)) minus 119891 (120593 (119905)) = 0 (8)


where

119886 (119905) = 11989611988831198982(1205932 (119905) + 11989821198882)32 (10)



2 (11)

with

12059021= 119896119898 (1 + ( 2119865

0119888119898120596)2)minus32

12059022= 119896119898

(12)



4119896 lt 1198981205962 (13)






11990910158401015840 + ℎ (119905) 119909 = 0 (15)

where

ℎ (119905) =

119891 (120593 (119905)) minus 119891 (120595 (119905))120593 (119905) minus 120595 (119905) if 120593 (119905) = 120595 (119905)

1198911015840 (120593 (119905)) otherwise(16)


0 lt ℎ (119905) lt 119896119898 lt (120587119879)

2

(17)








11990910158401015840 + 119886 (119905) 119909 + 119887 (119905) 1199092 + 119888 (119905) 1199093 + 119877 (119905 119909) = 0 (18)



120597119896119877120597119909119896 (119905 0) = 0 119896 = 0 1 2 3 forall119905 isin R (19)


11990910158401015840 + 119886 (119905) 119909 = 0 (20)



12058211205822= 1 (21)



12= plusmn1 and





the origin by

P (119909 119910) = 120593 (119879 119909 119910) (22)


120593 (0 119909 119910) = 1199091205931015840 (0 119909 119910) = 119910 (23)










0

119903 (119905)4 119888 (119905) 119889119905 +∬[0119879]2

119887 (119905) 119887 (119904) 1199033 (119905)sdot 1199033 (119904) 120594

2(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904

(25)


11991010158401015840 + 119886 (119905) 119910 = 11199103 (26)



8 sin (31205792) 119909 isin ]0 120579]

(27)






0 lt 12059021le 119886 (119905) le 120590

2

2 lt ( 1205872119879)2 forall119905 isin R (28)

119888lowastgt 10119879120590

2

7

3 sin ((31198792) 1205901) 12059081

119887lowast2 (29)


1205901211205902

le 119903 (119905) le 1205901221205901



119903 (119905) = 120572119903lowast (119905 minus 11990501205722 ) (31)



0+ 1205722120591) (32)



119894= 1205722120590



112057221205902

le 119903lowast le 112057221205901

(33)


11205721205902

le 119903 le 11205721205901

(34)


le 120572 le 120590minus121





1205901le 120588 le 120590

2lt 1205872119879 (35)


1205901119879 le 120579 le 120590

2119879 lt 120587

2 (36)



120573lowast ge 3811987911990341119888lowast +∬[0119879]2 (119887


811987911990341119888lowastminus∬[0119879]2


[0119879]2

119887minus (119905) 119887+ (119904)sdot 1199033 (119905) 1199033 (119904) 120594

2(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904

(37)



computed in [16]



18 sin ((31198792) 1205901) (39)



2 58 sin ((31198792) 120590

1) gt 0 (40)





11990910158401015840 + 119886 (119905) 119909 + 119887 (119905) 1199092 + c (119905) 1199093 + sdot sdot sdot = 0 (41)

where

119886 (119905) = 11989611988831198982(1205932 (119905) + 11989821198882)32

119887 (119905) = minus311989611989821198883120593 (119905)2 (1205932 (119905) + 11989821198882)52

119888 (119905) = 11989611989821198883 (41205932 (119905) minus 11989821198882)2 (1205932 (119905) + 11989821198882)72

(42)


1003816100381610038161003816120593 (119905)1003816100381610038161003816 lt 21198650120596 (43)




12059021= 119896119898 (1 + ( 2119865

0119888119898120596)2)minus32

12059022= 119896119898

119887lowast = 3119896119865012059611989831198882

119888lowast= 11989611989821198883 (11989821198882 minus 161198652

01205962)

2 (11989821198882 + 4119865201205962)72

(44)


Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0cos120596119905)119896



11990910158401015840 + 119891 (119909 + 120593 (119905)) minus 119891 (120593 (119905)) = 0 (8)


where

119886 (119905) = 11989611988831198982(1205932 (119905) + 11989821198882)32 (10)



2 (11)

with

12059021= 119896119898 (1 + ( 2119865

0119888119898120596)2)minus32

12059022= 119896119898

(12)



4119896 lt 1198981205962 (13)






11990910158401015840 + ℎ (119905) 119909 = 0 (15)

where

ℎ (119905) =

119891 (120593 (119905)) minus 119891 (120595 (119905))120593 (119905) minus 120595 (119905) if 120593 (119905) = 120595 (119905)

1198911015840 (120593 (119905)) otherwise(16)


0 lt ℎ (119905) lt 119896119898 lt (120587119879)

2

(17)








11990910158401015840 + 119886 (119905) 119909 + 119887 (119905) 1199092 + 119888 (119905) 1199093 + 119877 (119905 119909) = 0 (18)



120597119896119877120597119909119896 (119905 0) = 0 119896 = 0 1 2 3 forall119905 isin R (19)


11990910158401015840 + 119886 (119905) 119909 = 0 (20)



12058211205822= 1 (21)



12= plusmn1 and





the origin by

P (119909 119910) = 120593 (119879 119909 119910) (22)


120593 (0 119909 119910) = 1199091205931015840 (0 119909 119910) = 119910 (23)










0

119903 (119905)4 119888 (119905) 119889119905 +∬[0119879]2

119887 (119905) 119887 (119904) 1199033 (119905)sdot 1199033 (119904) 120594

2(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904

(25)


11991010158401015840 + 119886 (119905) 119910 = 11199103 (26)



8 sin (31205792) 119909 isin ]0 120579]

(27)






0 lt 12059021le 119886 (119905) le 120590

2

2 lt ( 1205872119879)2 forall119905 isin R (28)

119888lowastgt 10119879120590

2

7

3 sin ((31198792) 1205901) 12059081

119887lowast2 (29)


1205901211205902

le 119903 (119905) le 1205901221205901



119903 (119905) = 120572119903lowast (119905 minus 11990501205722 ) (31)



0+ 1205722120591) (32)



119894= 1205722120590



112057221205902

le 119903lowast le 112057221205901

(33)


11205721205902

le 119903 le 11205721205901

(34)


le 120572 le 120590minus121





1205901le 120588 le 120590

2lt 1205872119879 (35)


1205901119879 le 120579 le 120590

2119879 lt 120587

2 (36)



120573lowast ge 3811987911990341119888lowast +∬[0119879]2 (119887


811987911990341119888lowastminus∬[0119879]2


[0119879]2

119887minus (119905) 119887+ (119904)sdot 1199033 (119905) 1199033 (119904) 120594

2(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904

(37)



computed in [16]



18 sin ((31198792) 1205901) (39)



2 58 sin ((31198792) 120590

1) gt 0 (40)





11990910158401015840 + 119886 (119905) 119909 + 119887 (119905) 1199092 + c (119905) 1199093 + sdot sdot sdot = 0 (41)

where

119886 (119905) = 11989611988831198982(1205932 (119905) + 11989821198882)32

119887 (119905) = minus311989611989821198883120593 (119905)2 (1205932 (119905) + 11989821198882)52

119888 (119905) = 11989611989821198883 (41205932 (119905) minus 11989821198882)2 (1205932 (119905) + 11989821198882)72

(42)


1003816100381610038161003816120593 (119905)1003816100381610038161003816 lt 21198650120596 (43)




12059021= 119896119898 (1 + ( 2119865

0119888119898120596)2)minus32

12059022= 119896119898

119887lowast = 3119896119865012059611989831198882

119888lowast= 11989611989821198883 (11989821198882 minus 161198652

01205962)

2 (11989821198882 + 4119865201205962)72

(44)


Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119896119877120597119909119896 (119905 0) = 0 119896 = 0 1 2 3 forall119905 isin R (19)


11990910158401015840 + 119886 (119905) 119909 = 0 (20)



12058211205822= 1 (21)



12= plusmn1 and





the origin by

P (119909 119910) = 120593 (119879 119909 119910) (22)


120593 (0 119909 119910) = 1199091205931015840 (0 119909 119910) = 119910 (23)










0

119903 (119905)4 119888 (119905) 119889119905 +∬[0119879]2

119887 (119905) 119887 (119904) 1199033 (119905)sdot 1199033 (119904) 120594

2(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904

(25)


11991010158401015840 + 119886 (119905) 119910 = 11199103 (26)



8 sin (31205792) 119909 isin ]0 120579]

(27)






0 lt 12059021le 119886 (119905) le 120590

2

2 lt ( 1205872119879)2 forall119905 isin R (28)

119888lowastgt 10119879120590

2

7

3 sin ((31198792) 1205901) 12059081

119887lowast2 (29)


1205901211205902

le 119903 (119905) le 1205901221205901



119903 (119905) = 120572119903lowast (119905 minus 11990501205722 ) (31)



0+ 1205722120591) (32)



119894= 1205722120590



112057221205902

le 119903lowast le 112057221205901

(33)


11205721205902

le 119903 le 11205721205901

(34)


le 120572 le 120590minus121





1205901le 120588 le 120590

2lt 1205872119879 (35)


1205901119879 le 120579 le 120590

2119879 lt 120587

2 (36)



120573lowast ge 3811987911990341119888lowast +∬[0119879]2 (119887


811987911990341119888lowastminus∬[0119879]2


[0119879]2

119887minus (119905) 119887+ (119904)sdot 1199033 (119905) 1199033 (119904) 120594

2(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904

(37)



computed in [16]



18 sin ((31198792) 1205901) (39)



2 58 sin ((31198792) 120590

1) gt 0 (40)





11990910158401015840 + 119886 (119905) 119909 + 119887 (119905) 1199092 + c (119905) 1199093 + sdot sdot sdot = 0 (41)

where

119886 (119905) = 11989611988831198982(1205932 (119905) + 11989821198882)32

119887 (119905) = minus311989611989821198883120593 (119905)2 (1205932 (119905) + 11989821198882)52

119888 (119905) = 11989611989821198883 (41205932 (119905) minus 11989821198882)2 (1205932 (119905) + 11989821198882)72

(42)


1003816100381610038161003816120593 (119905)1003816100381610038161003816 lt 21198650120596 (43)




12059021= 119896119898 (1 + ( 2119865

0119888119898120596)2)minus32

12059022= 119896119898

119887lowast = 3119896119865012059611989831198882

119888lowast= 11989611989821198883 (11989821198882 minus 161198652

01205962)

2 (11989821198882 + 4119865201205962)72

(44)


Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894= 1205722120590



112057221205902

le 119903lowast le 112057221205901

(33)


11205721205902

le 119903 le 11205721205901

(34)


le 120572 le 120590minus121





1205901le 120588 le 120590

2lt 1205872119879 (35)


1205901119879 le 120579 le 120590

2119879 lt 120587

2 (36)



120573lowast ge 3811987911990341119888lowast +∬[0119879]2 (119887


811987911990341119888lowastminus∬[0119879]2


[0119879]2

119887minus (119905) 119887+ (119904)sdot 1199033 (119905) 1199033 (119904) 120594

2(1003816100381610038161003816120601 (119905) minus 120601 (119904)1003816100381610038161003816) 119889119905 119889119904

(37)



computed in [16]



18 sin ((31198792) 1205901) (39)



2 58 sin ((31198792) 120590

1) gt 0 (40)





11990910158401015840 + 119886 (119905) 119909 + 119887 (119905) 1199092 + c (119905) 1199093 + sdot sdot sdot = 0 (41)

where

119886 (119905) = 11989611988831198982(1205932 (119905) + 11989821198882)32

119887 (119905) = minus311989611989821198883120593 (119905)2 (1205932 (119905) + 11989821198882)52

119888 (119905) = 11989611989821198883 (41205932 (119905) minus 11989821198882)2 (1205932 (119905) + 11989821198882)72

(42)


1003816100381610038161003816120593 (119905)1003816100381610038161003816 lt 21198650120596 (43)




12059021= 119896119898 (1 + ( 2119865

0119888119898120596)2)minus32

12059022= 119896119898

119887lowast = 3119896119865012059611989831198882

119888lowast= 11989611989821198883 (11989821198882 minus 161198652

01205962)

2 (11989821198882 + 4119865201205962)72

(44)


Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











12059021= 119896119898 (1 + ( 2119865

0119888119898120596)2)minus32

12059022= 119896119898

119887lowast = 3119896119865012059611989831198882

119888lowast= 11989611989821198883 (11989821198882 minus 161198652

01205962)

2 (11989821198882 + 4119865201205962)72

(44)


Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Twist Periodic Solutions in the ...downloads.hindawi.com/journals/aaa/2016/6084082.pdfTwist Periodic Solutions in the Relativistic Driven Harmonic Oscillator ... .We