Nonlinear Analysis: Theory, Methods & Applications
On the period function of reversible quadratic centers with their orbits inside quartics☆
Introduction
Consider the planar polynomial system where are polynomials and the dot denotes differentiation with respect to time . 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 , transversal to them. Then the period function is defined as the period of the period orbit that passes through the point . A critical point of the period function means a value such that . It is possible to prove that the number of critical periods does not depend on the choice of . 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 where is a first integral of system (1.2) with integrating factor . Suppose that system (1.2) has at least one center. Then each periodic orbit surrounding the center is contained in a unique level set and its period equals
for .
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 , reversible , codimension four , generalized Lotka–Volterra systems and Hamiltonian triangle. Here we use the terminology from [5]. Taking a complex coordinate , the list of these five kinds of quadratic centers at can be looked upon as follows:
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 , then the system belongs to . Iliev classified into several cases. One of them, called the general case, has a first integral as follows
with the integrating factor , where . We refer the reader to [3] for more details.
For , the monotonicity of the period function has been studied by Zhao [13]. Moreover, if , then almost all orbits of 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 whose almost all orbits are quartic, provided that its first integral (1.4) with . In fact, we will consider the monotonicity of the period function of periodic trajectories for the following system: where is a real number with .
Now we change system (1.5) to a more convenient system.
By transformations system (1.5) reduces to
where .
We notice here that if , then system (1.5) is a Hamiltonian. Thus in the rest of this paper, we just deal with the cases where .
The following result has been obtained in [15]. The reader can also proof it by direct calculation.
Lemma 1.1 If , then system(1.6)has a center at , a saddle at , a stable node at and a unstable node at , respectively; If , then system(1.6)has a unique critical point at which is a center. If , then system(1.6)has two centers at and respectively; If , then system(1.6)has a unique critical point at which is a center. If , then system(1.6)has a center at and a saddle at respectively.
Our main result of this paper is the following theorem.
Theorem 1.1 Suppose that and that and are zeros of . If , then the period function of periodic orbits around the center is monotonous increasing. If , then the period function of periodic orbits around the center has a unique critical point, at which the period function admit a minimum if and a maximum if , respectively. If , then the period function of periodic orbits around the center is monotonous decreasing. If , then the period function of periodic orbits around the center is monotonous increasing. If , then the period function of periodic orbits around the center has a unique critical point, at which the period function admits a maximum. If , then the period function of periodic orbits around the center 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 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 is related with the first period constant so that and 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 and 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
It is easy to see from Lemma 1.1 that the center and their period trajectories always stay in the right half-plane. So we suppose in this situation that and take a change of coordinates to transform system (1.6) into
A first integral of system (2.2) is given by with integrating factor .
Now we give the domain of the corresponding period function for system (2.2)
Case 1. .
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 we have Define where the orientation of the integrals over the oval is clockwise.
It follows from (2.3) that which implies that where the prime ′ denotes the derivative with respect to .
Obviously,
Proof of main theorem
In this section, we denoted by the number of zeros of function with in the domain of (counting the multiplicities).
Lemma 4.1 If , then
Proof A straightforward computation yields that and , which imply that for if and that for if . Suppose that and are two consecutive zeros of , i.e., for , then we
Acknowledgements
The authors would like to thank the referees very much for their valuable comments and suggestions.
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Research partially supported by the NSF of China (No. 10871214) and Program for new century excellent talents in university.