Solution of Center-Focus Problem for a Class of Cubic Systems∗

Bo SANG Chuanze NIU

1 Introduction

Consider a two-dimensional system of differential equations of the form

wherePn,Qnare real polynomials of degreenwithout constant and linear terms.The singularity at the origin is a weak focus(surrounded by spirals)or a center(surrounded by closed trajectories).The Poincaré center-focus problem(see[1])is to determine conditions on the coefficients ofPnandQn,under which an open neighborhood of the origin is covered by closed trajectories of system(1.1).

Although the center-focus problem of system(1.1)has attracted intensive attentions,the characterization of centers for cubic systems is far from complete.Malkin[2]found necessary and sufficient conditions for a cubic vector field with no quadratic terms to have a center.For a cubic system of the form:

whereandare homogeneous polynomials of degrees,satisfyingChavarriga and Giné[3]gave a complete center characterization.

For a cubic system of the form:

Sadovskii and Shcheglova[4]presented a complete solution to the center-focus problem.For a special case of the above system withP=0,Hill,Lloyd and Pearson[5–7]obtained the necessary and sufficient conditions for the origin to be a center and an isochronous center,respectively.

A center of(1.1)is called to be isochronous if all cycles near it have the same period.It is well known that isochronous centers are non-degenerate.The problem to determine whether the center is isochronous or not is called the isochronicity problem.Although this problem has attracted the attentions of many authors,the characterization of isochronous centers even for cubic systems is far from complete.Recently,the isochronous center problem of time-reversible cubic systems was completely solved by Chen and Romanovski[8].For other developments of some polynomial differential systems,we refer to[9–15]and the references therein.

In the casen=3,system(1.1)can be written as

whereare homogeneous polynomials of degreek.

By the method of[16],there exists a unique formal power series of the form:

wherewithkeven andis a homogeneous polynomial of degreek,so that

whereWnis called thenth Liapunov constant of system(1.2).

The classical Poincaré-Liapunov method gives the usual version of de finition.There exists a unique formal power series of the form:

whereCk,k=0 withkeven andGk(x,y)is a homogeneous polynomial of degreek,so that

whereVnis also called thenth Liapunov constant of system(1.2).

In a general setting,the computational problems which appear in the computation of the Liapunov constants were discussed(see[16–18]).

Proposition 1.1For system(1.2)and any natural number n,V1=V2=···=Vn=0if and only if W1=W2=···=Wn=0.

ProofSuppose thatV1=V2=···=Vn=0,and then according to the work of Zhang et al.[17],we can uniquely determinesuch that

where

andPk−1=Qk−1=0 whenk≥5.

For any positive integermsatisfying 1≤m≤n,we set2m+2.From[16],it is known that the quantityWmis determined by the equation

and thusWm=0,1≤m≤n.

The converse direction can be proved in the same way.Suppose thatW1=W2=···=Wn=0,and then according to[16],we can uniquely determinesuch that

For any positive integermsatisfying 1≤m≤n,we set2m+2.From[17],it is known that the quantityVmis determined by the equation

and thusVm=0,1≤m≤n.

Corollary 1.1For system(1.2),the center varieties obtained from the polynomial ideals〈Wk:k≥1〉and〈Vk:k≥1〉are the same.

Proposition 1.2The nth Liapunov constant of system(1.2)is given by the formula

where

ProofLet

Then thenth Liapunov constant of system(1.2)is

So by the general Leibniz rule,we get the formula(1.5).

We remark that eachBi,jon the right hand side of(1.5)is a polynomial in the coefficients of system(1.2),and can be uniquely determined by using the identity(1.4).

In the casesystem(1.2)becomes

Applying Proposition 1.2 to system(1.6),we have the following corollary.

Corollary 1.2The nth Liapunov constant of system(1.6)is given by the formula

In Section 2,the computation of thenth Liapunov constants in the coefficients of cubic system(1.6)is considered.In Section 3,the solutions to the center-focus problem and the isochronous center problem for a particular case of system(1.6)are given.

2 The nth Liapunov Constant for System(1.6)and the Main Results

Let the time derivative ofHalong the orbits of system(1.6)be

wherefj,kare polynomials in the coefficients of system(1.6)and the functionH(x,y)in(1.3).

As a consequence of Corollary 1.2,for eachswe have

Suppose that(1.6)has a weak focus of ordernat the origin,and then the following conditions hold:

Inspired by the algorithm of Wang[16,19]and the corresponding Maple procedure licon[20]in Epsilon[miscel]package,we can represent thenth Liapunov constantWn,that is,in the coefficients of system(1.6)by succesive pseudo-divisions.

Letbe a polynomial set in the variablesx1,x2,···,xn,for which the main variable ofAjisxpj.The pseudo-remainder of a polynomialP(in the variableswith respect toAis

whereand prem(P,Ar,xpr)denotes the pseudo-remainder ofPmoduloArwith respect to the variablexpr.

Let

and

Set the first(n−1)Liapunov constants to be all zero,i.e.,for all 1≤k≤n−1 and letwherevis a dummy variable,so then thenth Liapunov constant can be obtained through a chain of pseudo-divisions:

The last equation gives thenth Liapunov constant in the coefficients of system(1.6),wherestands for the coefficient ofvin the polynomial

Although the basic idea of the previous formulae and the ones implicit in the Maple procedure licon(see[20])are the same,there are two differences:First,our formulae are designed just for system(1.6),while the Maple procedure licon is designed for the general polynomial differential systems;secondly,we find that it takes 2n2+4ntimes of pseudo-divisions to compute thenth order Liapunov constant by using our formulae,while in licon it takes 2n2+6ntimes.

Since the solution of the center-focus problem for system(1.6)is very difficult,we restrict our attention to a special system:

with

If,then using the scaling:,system(2.4)can be transformed into

in which

Ifc3,0＜0,then using the scaling:system(2.4)can be transformed into

in which

Considering the coefficient conditions(2.6)–(2.7)and(2.9)–(2.10),a discussion of(2.5)and(2.8)in the next section gives the following results.

Theorem 2.1For system(2.4)with c3,0/=0,the origin is a center if and only if one of the following conditions holds:

where k is a non-zero real parameter.

Theorem 2.2System(2.4)with cannot have an isochronous center at the origin.

3 Proof of Main Results

According to the recursive formulae in Section 2,we obtain the first seven Liapunov constants of system(2.5):

andW5,W6,W7have 66,105,64 terms,respectively.We would not present them here due to their lengthy expressions,but one can easily calculate them by using our formulae with the Maple computer algebra system.

Let,and then the idealJis called the Bautin ideal of system(2.5).And letbe the polynomial ideal generated byW1,W2,···,Wk.The affine varietyV(J)is called the center variety for the singular point at the origin of system(2.5).Computing a Gröbner basisGof the idealJ7with respect to the graded reverse lexicographical order with

we obtain a list of polynomials:

Before proving the main results,we start with four lemmas.

Lemma 3.1The center variety of system(2.5)is the variety of the ideal J7generated by the first seven Liapunov constants,and is composed of the following five components:

ProofUsing the radical ideal membership test,we can verify that

Thus we expect that.To verify it, first we find that

and then prove that every system fromVj,1≤j≤5 has a center at the origin.

Any system from the componentV1has the form

Since(3.1)is a Hamiltonian system,it has a center at the origin.

Any system from the componentV2has the form

By the transformation

system(3.2)becomes a time reversible system,and hence it has a center at the origin.

Any system from the componentV3has the form

which admits an integrating factor

so it has a center at the origin.

Any system from the componentV4has the form

which admits an integrating factor

so it has a center at the origin.

Any system from the componentV5has the form

which admits an integrating factor

and hence it has a center at the origin.

With the same principle,we have the following result about system(2.8).

Lemma 3.2The center variety of system(2.8)is the variety of the ideal K7generated by the first seven Liapunov constants,and is composed of the following five components:

Lemma 3.3System(2.5)cannot have an isochronous center at the origin.

ProofUsing the general algorithm in paper[9],we obtain the first two isochronous constantsp2,p4for system(2.5):

The simultaneous vanishing of polynomials ingives rise to only one case

whereGis the Gröbner basis ofJ7.For this case,(2.5)has the form

Again using the general algorithm in paper[9],we get the third-order isochronous constant of system(3.6):

which is positive for all realb2,0,and hence system(2.5)cannot have an isochronous center at the origin.

By the same method as in the proof of Lemma 3.3,we get the next lemma.

Lemma 3.4System(2.8)cannot have an isochronous center at the origin.

Proof of Theorem 2.1By Lemmas 3.1–3.2 and the coefficient conditions(2.6)–(2.7)and(2.9)–(2.10),we conclude that Theorem 2.1 holds.

Proof of Theorem 2.2By Lemmas 3.3–3.4 and the coefficient conditions(2.6)–(2.7)and(2.9)–(2.10),we conclude that Theorem 2.2 holds.

AcknowledgementsThe authors are grateful to Prof.Jaume Giné for his valuable remarks and comments.The authors are also grateful to the referees for their valuable remarks which helped to improve the manuscript.

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