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Generalized rational first integrals of analyticdifferential systems
Xiang Zhang
(Joint with Wang Cong and Jaume Llibre)
AQTDE2011, Castro Urdiales, Santander, Spain, Sep. 16
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Outline of the talk
Background of the problem.
Statement of the main results.
Sketch proof of the main results.
Some results on related problems.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Background of the problem
Given a Cω system of differential equations
x = f (x), x ∈Ω⊂ Rn an open set.
It is a classical problem:
to determine the existence of analytic or generalizedrational first integrals in Ω,
or
to determine the existence of analytic or generalizedrational first integrals in a neighborhood of some singularityin Ω.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Recall that
an analytic first integral is a first integral which is ananalytic function.
a generalized rational first integral is a first integral which isa ratio of two analytic functions.
Remark that
A generalized rational first integral with its denominatornonvanishing is an analytic first integrals .
A generalized rational first integral with its numerator anddenominator both polynomials is a rational first integral.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Remark:
This problem appears more than one hundred year,
but the progress is very slow.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
The history of the study on the problem:
Equivalent characterization of analytic first integrals (AFI):
planar nondegenerate centers having AFIs by Poincaré
planar nondegenerate isochronous centers having AFIs byPoincaré
planar hyperbolic saddles having AFIs by Moser [Comm.Pure Appl. Math. 1956]
planar nilpotent centers having AFIs by Chavarriga et al[Ergodic Theory Dynam. Systems 2003], partial results
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
any dimensional analytic differential systems aroundnondegenerate singularity which is analytic integrable byZhang [JDE 2008]
any dimensional analytic differential systems aroundsingularity with non–zero eigenvaluess which is analyticintegrable by Zhang [preprint]
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Rational first integrals (RFI):
the equivalent characterization, e.g
planar elementary singularities having a generalizedrational first integral by Li, Llibre and Zhang [BSM 2001]
the Darboux integrability
the existence of rational first integrals with sufficient numberof invariant algebraic surface (curves) by
Jouanolou [LNM 1979]Christopher and Llibre [ADE 2000, QTDS 1999]Llibre and Zhang [Bull Sci Math 2010, JDE 2009]
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Necessary conditions on existence of analytic first integrals
Consider analytic differential systems
x = Ax+ f (x), (1)
with f (x) = O(|x|2) analytic.
Letλ = (λ1, . . . ,λn)
be the n–tuple of eigenvalues of A.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Definition: the eigenvalues λ satisfy
Z+–resonant condition if
〈λ ,k〉= 0, for some k ∈(Z+)n, k 6= 0,
where Z+ is the set of nonnegative integers.
Z–resonant condition if
〈λ ,k〉= 0, for some k ∈ Zn, k 6= 0,
where Z is the set of integers.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Existence of analytic first integralsassociated with the eigenvalues of A:
Poincaré is the first one studying the relation between theexistence of analytic first integrals and resonances:
Poincaré TheoremAssume that
the eigenvalues λ of A do not satisfy any Z+–resonantconditions.
Then
system (1) has no analytic first integrals in (Cn,0).
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Remark:
λ do not satisfy resonant conditions implies that A has nozero eigenvalues.
The Poincaré’s result was extended by
Li, Llibre and Zhang [ZAMP 2003]
to the case that λ admits one zero eigenvalue.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Li, Llibre and Zhang’s Theorem [ZAMP, 2003]
Assume that
A has one zero eigenvalue, and
the others are not Z+–resonant.
Then
in the planar case, system (1) has an analytic first integralif and only if the origin is a non–isolated singularities
in the higher dimensional case, system (1) has a formalfirst integral if and only if the origin is a non–isolatedsingularities
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
In 2008, the Poincaré’s result was further extended to the caseof several first integrals:
Chen, Yi and Zhang’s Theorem [JDE, 2008]
The number of functionally independent analytic first integralsof system (1) does not exceed the maximal number of linearlyindependent elements of k ∈ (Z+)n : 〈k,λ 〉= 0, k 6= 0.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
In 2007, the Poincaré’s result was extended to the existence ofrational first integrals:
Shi’s Theorem [JMAA, 2007]
If system (1) has a rational first integral, then theeigenvalues λ of A satisfy a Z–resonant condition.
In other words, if λ do not satisfy any Z–resonantcondition, then system (1) has no rational first integrals in(Cn,0).
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Statement of our new results
Theorem 1Let λ = (λ1, . . . ,λn) be the eigenvalues of A. Then
the number of functionally independent generalizedrational first integrals of system (1) in (Cn,0) is at most thedimension of the minimal vector subspace of Rn containingthe set k ∈ Zn : 〈k,λ 〉= 0, k 6= 0.
This result was obtained by Cong, Llibre and Zhang in [JDE2011]
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Remark:
This result has improved
the Poincaré’s one
the Chen, Yi and Zhang’s one, and
the Shi’s one
by studying the existence of more than one functionallyindependent rational first integrals.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Remak:
The methods used in the proofof the Poincaré’s result,of the Chen, Yi and Zhang’s one, andof Shi’s one
are not enough to study the existence of more than onefunctionally independent generalized rational first integrals.
We will use a different approach to prove Theorem 1.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Preparation to the proof of Theorem 1
Let
C(x) be the field of rational functions in the variables x,
C[x] be the ring of polynomials in x.
Definition
functions F1(x), . . . ,Fk(x) ∈ C(x) are algebraicallydependent if there exists a complex polynomial P of kvariables such that P(F1(x), . . . ,Fk(x))≡ 0.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
The main steps of the proofs of Theorem 1
First step: algebraically and functionally independent
Lemma 1The functions F1(x), . . . ,Fk(x) ∈ C(x) are algebraicallyindependent if and only if they are functionally independent.
Remark
Lemma 1 has a relation in some sense with the result of Brunsin 1887, which stated that
if a polynomial differential system of dimension n hasl(1≤ l≤ n−1) independent algebraic first integrals, then ithas l independent rational first integrals.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
The main idea of the proof of Lemma 1
follows from that of Lemma 9.1 of Ito [Comment. Math.Helvetici, 1989].
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Sufficiency By contradiction,if F1(x), . . . ,Fk(x) are algebraically dependent,
⇓∃ a complex polynomial P(z1, . . . ,zk) of minimal degree such that
P(F1(x), . . . ,Fk(x))≡ 0.
minimal means that for any polynomial Q(z1, . . . ,zk) of degreeless than degP we have that Q(F1(x), . . . ,Fk(x)) 6≡ 0.
⇓
∂ (F1(x), . . . ,Fk(x))∂ (x1, . . . ,xn)
(∂P∂ z1
(F1, . . . ,Fk), . . . ,∂P∂ z1
(F1, . . . ,Fk)
)T
≡ 0.
⇓F1(x), . . . ,Fk(x) are functionally dependent
⇓contradiction with the assumptionXiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
NecessaryUsing the field extension and the expression of the derivativeson C(x).
1. F1, . . . ,Fk are algebraically independent,⇓
C(F1, . . . ,Fk) is a separably generated and finitely generatedfield extension of C of transcendence degree k
⇓there exist k derivations Dr(r = 1, . . . ,k) on C(F1, . . . ,Fk)
satisfyingDrFs = δrs,
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
2. Since C(x) is a finitely generated field extension ofC(F1, . . . ,Fk) of transcendence degree n− k
⇓there exist n derivations D1, . . . , Dn on C(x) satisfying
Dj = Dj on C(F1, . . . ,Fk) for j = 1, . . . ,k.
3. all derivations on C(x) form an n–dimensional vector spaceover C(x) with base ∂
∂xj: j = 1, . . . ,n
⇓
Ds =n
∑j=1
dsj∂
∂xj,
where dsj ∈ C(x).
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
4. The derivations Ds acting on C(F1, . . . ,Fk) satisfy
δsr = DsFr = DsFr =n
∑j=1
dsj∂Fr
∂xj, r,s ∈ 1, . . . ,k.
⇓the gradients ∇xF1, . . . ,∇xFk have the rank k,
⇓F1, . . . ,Fk are functionally independent.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Second step: independent in lowest degree terms
Notations
For an analytic or a polynomial function F(x) in (Cn,0),denote by F0(x) its lowest degree homogeneous term.
For a rational or a generalized rational functionF(x) = G(x)/H(x) in (Cn,0), denote by F0(x) the rationalfunction G0(x)/H0(x).
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
For analytic functions G(x) and H(x), expanding
F(x) =G(x)H(x)
=G0(x)H0(x)
+∞
∑i=1
Ai(x)Bi(x)
,
where Ai(x) and Bi(x) are homogeneous polynomials, and
degG0(x)−degH0(x)< degAi(x)−degBi(x) for all i≥ 1.
Definitions
degAi(x)−degBi(x) is called the degree of Ai(x)/Bi(x),
G0(x)/H0(x) is called the lowest degree term of F(x)
d(F) = degG0(x)−degH0(x) is called the lowest degree ofF.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Lemma 2Assume that
F1(x) =G1(x)H1(x)
, . . . ,Fm(x) =Gm(x)Hm(x)
,
are functionally independent generalized rational functionsin (Cn,0).
Then there exist polynomials Pi(z1, . . . ,zm) for i = 2, . . . ,m
such that
F1(x), F2(x) = P2(F1(x), . . . ,Fm(x)), . . . , Fm(x) =Pm(F1(x), . . . ,Fm(x)) are functionally independentgeneralized rational functions,
and F01(x), F
02(x), . . . , F
0m(x) are functionally independent
rational functions.Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Remark
Lemma 2 was first
proved by Ziglin [Functional Anal. Appl. 1983],
and then proved by Baider et al [Fields InstituteCommunications 7, 1996].
In our paper we also provide a proof using Lemma 1.The idea follows from the proof of Lemma 2.1 of Ito [Comment.Math. Helvetici 1989].
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Third step:
characterization of rational first integrals
Definition
A rational monomial is the ratio of two monomials, i.e. ofthe form xk/xl with k, l ∈ (Z+)n.
The rational monomial xk/xl is resonant if 〈λ ,k− l〉= 0.
A rational function is homogeneous if its denominator andnumerator are both homogeneous polynomials.
A rational homogeneous function is resonant if the ratio ofany two elements in the set of all its monomials in bothdenominator and numerator is a resonant rationalmonomial.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
The vector field associated to (1) is written in
X = X1 +Xh := 〈Ax,∂x〉+ 〈f (x),∂x〉.
Lemma 3If
F(x) = G(x)/H(x)
is a generalized rational first integral of the vector field X
defined by (1), then
F0(x) = G0(x)/H0(x)
is a resonant rational homogeneous first integral of the linearvector field X1.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Remark:
The proof of Lemma 3 needs the spectrum of linear operators
Define
Lc(h)(x) = 〈∂xh(x),Ax〉− ch(x), h(x) ∈H mn ,
where H mn the linear space of complex coefficient
homogeneous polynomials of degree m in n variables.
Then the spectrum of Lc is
〈k,λ 〉− c : k ∈ (Z+)n, |k|= k1 + . . .+ kn = m,
where λ are the eigenvalues of A.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Proof of Theorem 1
LetF1(x) =
G1(x)H1(x)
, . . . ,Fm(x) =Gm(x)Hm(x)
,
be the m functionally independent generalized rational firstintegrals of X .
⇓ by Lemma 2we can assume that
F01(x) =
G01(x)
H01(x)
, . . . ,F0m(x) =
G0m(x)
H0m(x)
,
are functionally independent.⇓ by Lemma 3
F01(x), . . . ,F
0m(x) are resonant rational homogeneous first
integrals of the linear vector field X1,Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
By the linear algebra, the matrix A in C has a uniquerepresentation in the form
A = As +An
with
As semi–simple, An nilpotent, AsAn = AnAs
As is similar to a diagonal matrix, and assume As diagonal
Define
Xs = 〈Asx,∂x〉 and
Xn = 〈Anx,∂x〉.
Separate X1 = Xs +Xn.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Direct calculations show that
any resonant rational monomial is a first integral of Xs,for example,
let xm be a resonant rational monomial, i.e.it satisfies 〈λ ,m〉= 0.
Then Xs(xm) = 〈λ ,m〉xm = 0.
So F01(x), . . . ,F
0m(x) are also first integrals of Xs.
This means that m is less than or equal to the number offunctionally independent resonant rational monomials.
In addition, the number of functionally independentresonant rational monomials is equal to the maximumnumber of linearly independent vectors in Rn of the setk ∈ Zn : 〈k,λ 〉= 0.
We complete the proof of the Theorem
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Related problem 1: semi-quasi-homogeneoussystems
Remark
If A of (1) has all its eigenvalues zero, Theorem 1 is trivial.
In this case, we consider semi–quasi–homogeneoussystems.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Systemx = f (x) = (f1(x), . . . , fn(x)), (2)
is quasi–homogeneous of degree q ∈ N\1 withexponents s = (s1, . . . ,sn) ∈ Zn \0 if for all ρ > 0
fi (ρs1x1, . . . ,ρsnxn) = ρ
q+si−1fi(x1, . . . ,xn), i = 1, . . . ,n.
is semi–quasi–homogeneous of degree q with the weightexponent s if
f (x) = fq(x)+ fh(x),
withx = fq(x),
quasi-homogeneous and fh(x) consisting of higher orderterms.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
SetW = diag(s1/(q−1), . . . ,sn/(q−1)).
Any solution c = (c1, . . . ,cn) of
fq(c)+Wc = 0,
is called a balance.
For each balance c, the eigenvalues of
K = Dfq(c)+W
are called Kowalevskaya exponents, denoted by λc.
Let dc be the dimension of the minimal vector subspace ofRn containing the set
k ∈ Zn : 〈k,λc〉= 0, k 6= 0 .
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Theorem 2Assume that
system (2) is semi–quasi–homogeneous of weight degreeq with weight exponent s.
Then
the number of functionally independent generalizedrational first integrals of (2) is at most d = min
c∈Bdc,
where B is the set of balances.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Remark
Theorem 2 is an extensionof Theorem 1 of Furta [ZAMP 1996]of Corollary 3.7 of Goriely [JMP 1996] andof Theorem 2 of Shi [JMAA 2007].
In some sense Theorem 2 is also an extensionof the main results of Yoshida [Celestial Mech 1983],
where he proved that if a quasi–homogenous differentialsystem is algebraically integrable, then every Kowalevkayaexponent should be a rational number.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Related problem 2: around periodic orbits
Remark:
Theorems 1 and 2 studied the existence of functionallyindependent generalized rational first integrals in aneighborhood of a singularity.
Next we investigate the existence of generalized rational firstintegrals of system (2) in a neighborhood of a periodic orbit.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Definition
The multipliers of a periodic orbit are the eigenvalues ofthe linear part of the Poincaré map at the fixed pointcorresponding to the periodic orbit.
Recall that
Associated to Poincaré map of a periodic orbit, its linearpart has the eigenvalue 1 along the direction tangent to theperiodic orbit.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Theorem 3Assume that
the analytic differential system (2) has a periodic orbit withmultipliers µ = (µ1, . . . ,µn−1).
Then
the number of functionally independent generalizedrational first integrals of system (2) in a neighborhood ofthe periodic orbit is at most the maximum number oflinearly independent vectors in Rn of the set
k ∈ Zn−1 : µk = 1, k 6= 0.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Related problem 3: periodic differential systems
Remark:
Previously, we studied the existence of functionally independentgeneralized rational first integrals of autonomous differentialsystems.
Finally we consider the periodic differential systems
x = f (t,x), (t,x) ∈ S1× (Cn,0), (3)
where
S1 = R/(2πN), and
f (t,x) is analytic in its variables and 2π periodic in t.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Definition
A non–constant function F(t,x) is a generalized rationalfirst integral of system (3) if
F(t,x) = G(t,x)/H(t,x) with G(t,x) and H(t,x) analytic in theirvariables and 2π periodic in t,and it satisfies
∂F(t,x)∂ t
+ 〈∂xF(t,x), f (t,x)〉 ≡ 0 in S1× (Cn,0).
functions F1(t,x), . . . ,Fm(t,x) are functionally independent inS1× (Cn,0) if
∂xF1(t,x), . . . ,∂xFm(t,x) have the rank m in a full Lebesguemeasure subset of S1× (Cn,0).
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Assume that
x = 0 is a constant solution of (3), i.e. f (t,0) = 0.
then system (3) can be written in
x = A(t)x+g(t,x), (4)
where A(t) and g(t,x) = O(x2) are 2π periodic in t.
Let
L be the monodromy operator associated with the linearperiodic equation
x = A(t)x.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems
Theorem 4Assume that
the monodromy operator L has the eigenvaluesµ = (µ1, . . . ,µn).
Then
the number of functionally independent generalizedrational first integrals of system (4) in a neighborhood ofthe constant solution x = 0 is at most the maximum numberof linearly independent vectors in Rn of the set
Ξ :=
k ∈ Zn : µk = 1,k 6= 0
⊂ Zn.
Xiang Zhang: Shanghai Jiaotong University, Shanghai Generalized rational first integrals of analytic systems