Limit Cycles Bifurcating from a Quadratic Reversible Lotka-Volterra System with a Center and Three Saddles?

Kuilin WU Haihua LIANG

1 Introduction

The second part of Hilbert’s 16th problem(see[6])asks about the maximum number and the location of limit cycles of a planar polynomial vector fields of degreen.If the quadratic centers belong to the Hamiltonian class,then the study of the number of limit cycles bifurcating from a period annulus or annuli(i.e.,the weak Hilbert’s 16th problem forn=2)is finished,and the study of the number of limit cycles bifurcating from a singular loop,or from infinity is partially finished(see[3–4,8–9,12,15,17]and the references therein).If the quadratic centers belong to the reversible class(and do not belong to the Hamiltonian class),then the study seems very difficult,and the known results are very limited.

A weaker version of this problem is proposed by Arnold(see[1])to study the zeros of Abelian integrals,that is the weak Hilbert’s 16th problem or infinitesimal Hilbert’s 16th problem.The problem is related to in the following way.Consider the perturbed system of a Hamiltonian vector fieldXε=XH+εY,where

A closed connected component of a level curve{H=h}is denoted byγhand called an oval ofH.The Abelian integral is

Therefore the number ofisolated zeros ofI(h),counted with multiplicities,provides an upper bound for the number of ovals of{H=h}that generates limit cycles ofXεforε≈0(see[5,12]for details).If the unperturbed system is integrable and non-Hamiltonian,then one has to consider pseudo-Abelian integrals(see[1]for details).As far as we know,most of the papers investigate the Hamiltonian centers and few papers study the non-Hamiltonian centers(see[3,8,17]for instance).Recently,Zhao[16]proved that the cyclicity of the period annulus ofQ4is less than or equal to five.

It is well-known by Iliev[8]that any quadratic polynomial reversible Lotka-Volterra system can be written in the complex form

or in the real form

wherebis a real parameter.Arnold[2]declared that the in finitesimal problem for system(1.1)is still open.For the system(1.1)of genus one,Gautier et al.[4]classify this kind of systems into 6 cases(rlv1)–(rlv6).Until now,[7,11]have studied the cases(rlv1)and(rlv2),respectively,and[5,14]have studied the cases(rlv3)and(rlv4),respectively.

The cyclicity of system(1.1)under quadratic perturbations forb=0,,was studied in[3–4],respectively.In this paper we study the caseb=,that is,the number of limit cycles bifurcates from the period annulus of the following quadratic reversible Lotka-Volterra system:

System(1.2)has a first integral

with the integrating factor

Note that system(1.1)forb=can be reduced to system(1.2)by using a linear transformation.There is a period annulus surrounding the center at(x,y)=(0,0)bounded by three straight lines

corresponding toThe intersection pointsandof the three straight lines are three saddles of systemTherefore,the period annulus can be expressed bywhere the periodic orbith}.

The next result is a particular case of Theorem 3 in[8].For convenience,we state it in the present paper.

Lemma 1.1The exact upper bound for the number of limit cycles produced by the period annulus of system(1.2)under quadratic perturbations is equal to the maximal number of zerosin(counting multiplicities)of the Abelian integral

where a,b and c are arbitrary constant.

The main result of this paper is the following theorems.

Theorem 1.1The cyclicity of the period annulus of system(1.2)under quadratic perturbations is two.

This paper is organized in the following way.In Section 2 we introduce the definitions and the notations that we shall use.In Section 3 we give the proof of Theorem 1.1 by applying Theorem B in[5].We shall rewrite the Abelian integral(1.3)as a linear combination of{I0(h),I1(h),I2(h)}and prove that(I0(h),I1(h),I2(h))forms an extended complete Chebychev system.Due to Theorem B in[5],we turn the problem of the number of zeros of the Abelian integral into a pure algebraic problem,namely,counting zeros of a polynomial.To solve the latter problem we shall use the notion of a resultant.The interested reader is referred to the appendix in[5]for details.

2 The Definitions of Chebyshev Systems

In order to prove the main results,some definitions and lemmas are needed.

Definition 2.1Let f0(x),f1(x),···,fn?1(x)be analytic functions on an open interval L ofR.Then we have the following:

(a)(f0(x),f1(x),···,fn?1(x))is a Chebyshev system(in short,T-system)on L if any nontrivial linear combination

has at most n?1isolated zeros on L.

(b)(f0(x),f1(x),···,fn?1(x))is a complete Chebyshev system(in short,CT-system)on L if(f0(x),f1(x),···,fk?1(x))is a T-system for all k=1,2,···,n.

(c)(f0(x),f1(x),···,fn?1(x))is an extended complete Chebyshev system(in short,ECT-system)on L if,for all k=1,2,···,n,any nontrivial linear combination

has at most k?1isolated zeros on L counted with multiplicities.

It is clear that if(f0(x),f1(x),···,fn?1(x))is an ECT-system onL,then(f0(x),f1(x),···,fn?1(x))is a CT-system onL.However,the reverse implication is not true.Moreover,if(f0(x),f1(x),···,fn?1(x))is a T-system onL,andf(x)is an analytic function and has a constant sign onL,then(f(x)f0(x),f(x)f1(x),···,f(x)fn?1(x))is a T-system onL.

Remark 2.1If(f0(x),f1(x),···,fn?1(x))is an ECT-system onL,then,for eachk=0,1,2,···,n?1,there exists a linear combination with exactlyksimple zeros onL(see[10,13]for instance).

Definition 2.2Let f0(x),f1(x),···,fn?1(x)be analytic functions on an open interval L of.The Wronskian of(f0(x),f1(x),···,fn?1(x))at x∈L is given by

The following result is well-known(see[10,13]for instance).

Lemma 2.1The system(f0(x),f1(x),···,fn?1(x))is an ECT-system on L if and only if,for each k=1,2,···,n,

3 Proof of Main Results

Recall thatH(x,y)=A(x)+B(x)y2with

It is clear thatH(x,y)has a local minimum at(x,y)=(0,0),B(x)>0 andA(x)have a local minimum atx=0.Denote the period ann(ulus ass)ociated to the center origin byand the projection ofon thex-axis byThen there exists a unique analytic involution functionσ(x),such thatfor allThe next result is a particular case of Theorem B in[5].For convenience,we state it in the present paper.

Theorem 3.1(see[5])Let us consider the Abelian integrals

where,for each h∈(0,h0),γhis the oval surrounding the origin inside the level curve{A(x)+B(x)y2=h}.Let σ be the involution associated to A(x)and we define

Then(I0,I1,···,In?1)is an ECT-system on(0,h0)if s>n?2and(l0,l1,···,ln?1)is a CT-system on(0,xr).

Recall that a mappingσis an involution ifσ?σ=Id andσ≠Id.An involutionσis a diffeomorphism with a unique fixed point.Noting thatli(x)=?li(σ(x)),we have that(l0,l1,···,ln?1)is a CT-system on(0,xr)if and only if(l0,l1,···,ln?1)is a CT-system on(xl,0).

We rewrite the Abelian integral(1.3)asI(h)=aI0(h)+bI1(h)+cI2(h),where

The following lemma,proved in[5],establishes a formula to change the integrand of an Abelian integral into other Abelian integrals that we want.

Lemma 3.1(see[5])Let γhbe an oval inside the level curve{A(x)+B(x)y2=h},and consider a function F(x)such thatis analytic at x=0.Then,for any k∈N,

where

In what follows,we shall apply Theorem 3.1 to prove that(I0(h),I1(h),I2(h))is an ECT-system.By Lemma 3.1,it yields that

However,we discover thatn=3 ands=1 in the integrand of(I0(h),I1(h),I2(h)),so that the conditions>n?2 is not fulfilled.Therefore we must takes=3 and apply Lemma 3.1 to overcome the shortcomings.Applying Lemma 3.1,we have that

where

Exactly in the same way we obtain

where

It is clear that(I0(h),I1(h),I2(h))is an ECT-system in the intervalif and only ifis an ECT-system in the interval

Setting

then

Denote byσthe involution associated toA(x),i.e.,A(x)=A(σ(x)).In order to compute Wronskians,we setz=σ(x)and

Thenli(x)=Li(x,z).We only need to prove that system(L0(x,z),L1(x,z),L0(x,z))is an ECT-system onOn account of

it turns out thatz=σ(x)is defined by

and

We shall depend on Wolfram Mathematica to compute three Wronskians and the resultant between two polynomials to show the nonexistence of zeros of a polynomial on the interval.In the following,we show the following lemma.

Lemma 3.2Systemis an ECT-system on the intervali.e.,systemis an ECT-system on the intervalwhere z is defined by(3.2).

ProofBy Lemma 2.1,we split the proof into three cases to show that the three Wronskians have no zeros on

First,note thatW[L0(x,z)]=L0(x,z).By the common denominator and the factorization,we have

whereα0(x,z)is a polynomial of degree 15 in(x,z)with a very long expression.It follows from direct computations that the resultant with respect tozbetweenα0(x,z)andq(x,z)is(i.e.,eliminatingzfromα0(x,z)=0 andq(x,z)=0),where

Note that=36,andhas a local minimum 35.0652 atx≈0.486258 on the interval.Therefore,α0(x,z)=0 andq(x,z)=0 have no common roots for any,which implies thatW[L0(x,z)]≠0 for anyorz∈(?1,0).

Secondly,by the definition ofW[L0(x,z),L1(x,z)],it follows that

whereα1(x,z)is a polynomial of degree 20 in(x,z)with a very long expression.It follows from direct computations that the resultant with respect tozbetweenα1(x,z)andq(x,z)is 131072(1+x)22(?1+2x)10ζ1(z),where

By calculating,we have thatζ1(0)=2799360,=6561,andζ1(x)has a local minimum 6128.2996 atx≈0.487723 on the intervalTherefore,α1(x,z)=0 andq(x,z)=0 have no common roots on the interval,andW[L0(x,z),L1(x,z)]≠0 for any

Finally,let us compute the third Wronskian and we have that

whereα2(x,z)is a polynomial of degree 50 in(x,z)with a very long expression.The resultant with respect tozbetweenα2(x,z)andq(x,z)is 618475290624(1+x)37(?1+2x)19ζ2(x),where

ζ2(0)=314424115200,=76527504,andhas a local minimum 7.07214×107atx≈0.490262 on the intervalHence,we can assert thatW[L0,L1,L2]≠0 for anyBy Lemma 2.1,the proof of the result is completed.

Proof of Theorem 1.1By Lemma 1.1,Lemma 3.2 and Theorem 3.1,we obtain Theorem 1.1.

Remark 3.1The proof depends on the symbolic computations by Wolfram mathematica and some very long expressions are omitted for the sake of briefness,while the derivative process can be done precisely.

[1]Arnold,V.I.,Some unsolved problems in the theory of differential equations and mathematical physics,Russian Math.Surveys,44,1989,157–171.

[2]Arnold,V.I.,Arnold’s Problems,Springer-Verlag,Berlin,2005.

[3]Chicone,C.and Jacobs,M.,Bifurcation of limit cycles from quadratic isochronous,J.Differential Equations,91,1991,268–326.

[4]Gautier,S.,Gavrilov,L.and Iliev,I.D.,Perturbations of quadratic centers of genus one,Discrete Contin.Dyn.Syst.,25,2009,511–535.

[5]Grau,M.,Ma?osas,F.and Villadelprat,J.,A Chebyshev criterion for Abelian integrals,Trans.Amer.Math.Soc.,363,2011,109–129.

[6]Hilbert,D.,Mathematische Problem(Lecture),Second Internat.Congress Math.Paris 1900,Nachr.Ges.Wiss.G¨ottingen Math.-Phys.Kl.,1900,253–297.

[7]Iliev,I.D.,The cyclicity of the period annulus of the quadratic Hamiltonian triangle,J.Differential Equations,128,1996,309–326.

[8]Iliev,I.D.,Perturbations of quadratic centers,Bull.Sci.Math.,122,1998,107–161.

[9]Ilyashenko,Y.,Centennial history of Hilbert’s 16th problem,Bull.Amer.Math.Soc.,39,2002,301–354.

[10]Karlin,S.and Studden,W.,TchebycheffSystems:With Applications in Analysis and Statistics,Interscience Publishers,New York,1966.

[11]Li,C.and Llibre,J.,The cyclicity of period annulus of a quadratic reversible Lotka-Volterra system,Nonlinearity,22,2009,2971–2979.

[12]Li,J.,Hilbert’s 16th problem and bifurcations of planar polynomial vector fields,Internat.J.Bifur.Chaos Appl.Sci.Engrg.,13,2003,47–106.

[13]Marde?i?,P.,Chebyshev Systems and the Versal Unfolding of the Cusp of Ordern,Travaux en Cours,57,Hermann,Paris,1998.

[14]Shao,Y.and Zhao,Y.,The cyclicity and period function of a class of quadratic reversible Lotka-Volterra system of genus one,J.Math.Anal.Appl.,377,2011,817–827.

[15]Zhao,Y.and Zhang,Z.,Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians,J.Differential Equations,155,1999,73–88.

[16]Zhao,Y.,On the number of limit cycles in quadratic perturbations of quadratic codimension-four centres,Nonlinearity,24,2011,2505–2522.

[17]Zoladek,H.,Quadratic systems with centers and their perturbations,J.Differential Equations,109,1994,223–273.