On the period function of reversible quadratic centers with their orbits inside quartics

https://doi.org/10.1016/j.na.2009.04.062Get rights and content

Abstract

This paper is concerned with the monotonicity of the period function for a class of reversible quadratic centers with their orbits inside quartics. It is proved that such a system has a period function with at most one critical point.

Introduction

Consider the planar polynomial system ẋ=p(x,y),ẏ=q(x,y), where p(x,y),q(x,y) are polynomials and the dot denotes differentiation with respect to time t. Suppose that system (1.1) has at least one center. It is well known that a continuum of periodic orbits surrounding a center can be parameterized by a curve γ{γ(h)|h(h0,h1)}, transversal to them. Then the period function T(h) is defined as the period of the period orbit that passes through the point γ(h). A critical point of the period function means a value h(h0,h1) such that T(h)=0. It is possible to prove that the number of critical periods does not depend on the choice of γ(h). Recall that the behavior of the period function sometimes plays an important role in the study of Abelian integrals and nonlinear boundary value problem, see for instance [1], [2], [3], [4].

In this paper we deal with the period function of the quadratic centers. We consider the planar quadratic system ẋ=p2(x,y)=Hy(x,y)/M(x,y),ẏ=q2(x,y)=Hx(x,y)/M(x,y), where H(x,y) is a first integral of system (1.2) with integrating factor M(x,y). Suppose that system (1.2) has at least one center. Then each periodic orbit Γh surrounding the center is contained in a unique level set {(x,y):H(x,y)=h} and its period equals

T(h)=Γhdt=Γhdxp2(x,y) for h(h0,h1).

The period function for (1.2) has been studied by a number of authors. In the literature different terminologies are used to classify quadratic systems with centers but essentially there are five families: Hamiltonian Q3H, reversible Q3R, codimension four Q4, generalized Lotka–Volterra systems Q3LV and Hamiltonian triangle. Here we use the terminology from [5]. Taking a complex coordinate z=x+iy, the list of these five kinds of quadratic centers at z=0 can be looked upon as follows: ż=izz2+2|z|2+(b+ic)z¯2,Hamiltonian (Q3H)ż=iz+az2+2|z|2+bz¯2,reversible(Q3R)ż=iz+4z2+2|z|2+(b+ic)z¯2,|b+ic|=2,codimension four (Q4)ż=iz+z2+(b+ic)z¯2,generalized Lotka–Volterra(Q3LV)ż=iz+z¯2,Hamiltonian triangle

Copel and Gavrilov [6] prove that the period functions of any Hamiltonian quadratic center are monotonous. Recently, Zhao [7] showed that the codimension four centers have the same property. Concerning the Lotka–Volterra systems there are very few results. In the middle 80s several authors [8], [4], [9] showed independently the monotonicity of the classical Lotka–Volterra center. Very recently, J. Villadelprat [10] showed that the period function is globally (i.e., in the whole period annulus) monotone increasing in two other cases different from the classical one.

Generally, The period function of the quadratic reversible centers is not monotone. So, from the point of view of the study of the period function it is clear that the most interesting family of centers is the reversible one. For more details, please see [11], [12], [13] and the references therein.

If b1, then the system ż=iz+az2+2|z|2+bz̄2 belongs to Q3R/Q3H. Iliev classified Q3R/Q3H into several cases. One of them, called the general case, has a first integral as follows

H(X,y)=Xa+b+2ab[12y2+18(ab)2(a+b2a3b2X2+2b1b+1X+a3b+2a+b+2)] with the integrating factor M(X,y)=X2a+2ab, where X=1+2(ab)x,b1. We refer the reader to [3] for more details.

For 2a+2ab=4, the monotonicity of the period function has been studied by Zhao [13]. Moreover, if 2a+2ab=3, then almost all orbits of Q3R/Q3H are cubic, the period function of such a system was investigated in the paper [14].

In the present paper, we take one of the general cases of system Q3R/Q3H whose almost all orbits are quartic, provided that its first integral (1.4) with a+b+2ab=32. In fact, we will consider the monotonicity of the period function of periodic trajectories for the following system: ẋ=y+8(b+1)xy,ẏ=x2(3b+1)x2+6(b+1)y2, where b is a real number with b1.

Now we change system (1.5) to a more convenient system.

By transformations x1+8(b+1)x,y82(b+1)y,dt12dt, system (1.5) reduces to

ẋ=xy,ẏ=2A2+(A2)xA2x2+34y2, where A=1+3bb+1.

We notice here that if b=1, then system (1.5) is a Hamiltonian. Thus in the rest of this paper, we just deal with the cases where A=1+3bb+13.

The following result has been obtained in [15]. The reader can also proof it by direct calculation.

Lemma 1.1

  • (1)

    If A>4 , then system(1.6)has a center at P1(1,0) , a saddle at P2(A4A,0) , a stable node at P3(0,23(A4)) and a unstable node at P4(0,23(A4)) , respectively;

  • (2)

    If A=4 , then system(1.6)has a unique critical point at P1(1,0) which is a center.

  • (3)

    If A(0,3)(3,4) , then system(1.6)has two centers at P1(1,0) and P2(A4A,0) respectively;

  • (4)

    If A=0 , then system(1.6)has a unique critical point at P1(1,0) which is a center.

  • (5)

    If A<0 , then system(1.6)has a center at P1(1,0) and a saddle at P2(A4A,0) respectively.

Our main result of this paper is the following theorem.

Theorem 1.1

Suppose that A3 and that A1=13249100.277973 and A2=13+249102.877973 are zeros of 5A213A4 .

  • (1)

    If A(,A1][A2,+) , then the period function of periodic orbits around the center P1(1,0) is monotonous increasing. If A(A1,0)(2,A2) , then the period function of periodic orbits around the center P1(1,0) has a unique critical point, at which the period function admit a minimum if A(A1,0) and a maximum if A(2,A2) , respectively. If A[0,2] , then the period function of periodic orbits around the center P1(1,0) is monotonous decreasing.

  • (2)

    If A(0,4A2] , then the period function of periodic orbits around the center P2(A4A,0) is monotonous increasing. If A(4A2,2) , then the period function of periodic orbits around the center P2(A4A,0) has a unique critical point, at which the period function admits a maximum. If A[2,4) , then the period function of periodic orbits around the center P2(A4A,0) is monotonous decreasing.

It follows from this theorem that

Corollary 1.1

For system(1.5), the corresponding period function has at most one critical point.

Remark

Compactifying R2 to the Poincaré disc, the boundary of the period annulus of the center has two connected components, the center itself and a polycycle. In [11], the authors call them respectively the inner and outer boundaries of the period annulus. In the proof of Theorem 1.1 (see Section 4), we can easy see that the function 5A213A4 is related with the first period constant so that A=A1 and A=A2 are the parameters where the period function undergoes a bifurcation at the inner boundary. (For more details, please see Lemma 3.6 or Lemma 4.3, Lemma 4.6). The parameters A=0 and A=2 are the bifurcation values of the period function at the outer boundary (see Lemma 4.3, Lemma 4.5). The precise definition of the bifurcation value of the period function can be found in the Definition 2.4 of [11].

In the literature many different methods are being applied to study the period function, see [16], [17], [18], [19], [20], [6], [21], [22], [23], [24], [11], [12], [9], [13] and the references therein. The idea used in the present paper is based on Picard–Fuchs equation for algebraic curves. We notice that Chicone [25] conjectured that the period function of the quadratic reversible centers have at most two critical points. So the content of this paper can also be regarded as a contribution to the proof of Chicone’s conjecture.

Section snippets

Determine the domain for the period function T(h)

It is easy to see from Lemma 1.1 that the center P1(1,0) and their period trajectories always stay in the right half-plane. So we suppose in this situation that x>0 and take a change of coordinates xx,yy,dt12dt to transform system (1.6) into ẋ=xy,ẏ=(A4)+2(A2)x2Ax4+32y2.

A first integral of system (2.2) is given by H(x,y)=x3(12y2+Ax4+2(A2)x2A43)=h with integrating factor μ(x,y)=x4.

Now we give the domain of the corresponding period function for system (2.2)

Case 1. A>4.

System (2.2)

Picard–Fuchs equation and some preliminary results

In this section, we will express the period function as a derivative of a suitable Abelian integral and derive some differential equations related to period function. Since dxdt=xy, we have T(h)=Γhdt=Γhdxxy. Define Ji(h)=Γhμ(x,y)xiydx=Γhxi4ydx,i=,1,0,1,, where the orientation of the integrals over the oval Γh is clockwise.

It follows from (2.3) that yh=x3y, which implies that Ji(h)=Γhxi1ydx,i=,1,0,1,, where the prime ′ denotes the derivative with respect to h.

Obviously, T(h)=J0(h

Proof of main theorem

In this section, we denoted by f(h) the number of zeros of function f(h) with h in the domain of f (counting the multiplicities).

Lemma 4.1

If A(,0)[2,4)(4,+) , thenϕ(h)R0(h)+1.

Proof

A straightforward computation yields that a21(h1)=128(A1)(A4),a21(h2)=128A(A3) and a21(h)=18(A2)(A4)h, which imply that a21(h)0 for h(h1,h2) if A(,0)(4,+) and that a21(h)0 for h(h1,) if A[2,4).

Suppose that h1˜ and h2˜ are two consecutive zeros of ϕ(h), i.e., ϕ(h1˜)=ϕ(h2˜)=0,ϕ(h)0 for h1˜<h<h2˜, then we

Acknowledgements

The authors would like to thank the referees very much for their valuable comments and suggestions.

References (29)

  • S.N. Chow et al.

    On the number of critical points of period

    J. Differential Equations

    (1986)
  • A. Cima et al.

    On polynomial Hamiltonian planar vector fields

    J. Differential Equations

    (1993)
  • A. Cima et al.

    Isochronicity for several classes of Hamiltonian systems

    J. Differential Equations

    (1999)
  • A. Gasull et al.

    The period function for Hamiltonian system with homogeneous nonlinearities

    J. Differential Equations

    (1997)
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    Research partially supported by the NSF of China (No. 10871214) and Program for new century excellent talents in university.

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