Generalized rational first integrals of analytic differential

Generalized rational first integrals of analyticdifferential systems

Xiang Zhang

(Joint with Wang Cong and Jaume Llibre)

[email protected]

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.


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.


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.


Remark:

This problem appears more than one hundred year,

but the progress is very slow.


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


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]


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]


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.


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.


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).


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.


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


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.


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).


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]


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.


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.


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.


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.


The main idea of the proof of Lemma 1

follows from that of Lemma 9.1 of Ito [Comment. Math.Helvetici, 1989].


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,


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).


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.


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).


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.


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].


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.


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.


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.


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.


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


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.


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.


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 .


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.


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.


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.


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.



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.


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.


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).


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.



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.


!

Gracias!

Thanks for your attention!


Generalized rational first integrals of analytic differential

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