ON THE MONOTONICITY OF THE PERIOD FUNCTION OF REVERSIBLE CENTERS∗

In this paper we study the period function of centers for a class of reversible systems and give a criteria to determine the monotonicity of the period functions.


Introduction and main results
It is well known that the center is surrounded by a continuous set of periodic orbits, which is called the period annulus of the center and denoted by P. The period function is the period corresponding to the periodic orbits in P. The center is called an isochronous center if the period function is a constant (see [1]). The critical periods are the zeros of the derivative of period function. It can be shown that the number of the critical periods does not depend on the particular parametrization used (see [6,7]). A system with a center is said to be reversible if its orbits are symmetric with respect to a straight line passing through the center. In this paper we study the monotonicity of period functions of a class of reversible centers. Consider the reversible systems: {ẋ = −U (x)y, y = f (x, y). (1.1) Suppose that (1.1) has a first integral of the form H(x, y) = F (x)y 2 + G(x). Then Denote by (x I , x S ) is the projection of the period annulus P to the x−axis, i.e.
Assume that F (x), G(x), U (x) are analytic functions on (x I , x S ). It is easy to verify that the origin is a center and M (x) = 2F (x) U (x) is an integral factor of system (1.1). Without loss of generality, we always assume that (H1) : The origin is a nondegenerate centers of (1.1), and U (0) > 0.
Denote the range of H on P by (0, h 0 ), where {H = h 0 } is corresponding to the boundary of the period annulus P with h 0 ≤ ∞. From (H1), it is easy to obtain that For every h ∈ (0, h 0 ), let us denote the periodic orbit of P corresponding to {H = h} by γ h and denote by T (h) its period. Moreover, we define Recall that an analytic diffeomorphism σ is said to be an involution if σ • σ = Id In fact, we may take σ( . In this paper, we obtain the following results for the reversible system (1.1). Theorem 1.1. Assume that the origin is a nondegenerate center of system (1.1) and that the function T (h) is the period function of the periodic orbit γ h . We take If S σ (µ k )(x) > 0 (or < 0) for x ∈ (0, x S ), then the period function of system (1.1) is monotone.
Recall that the behavior of the period function plays an important role in the study of Abelian integrals (see [2,8] for instance). Moreover it is also important in the study of other dynamical problems (see [3,4]). Over the years the problem for the period function have been extensively studied. F. Mañsas and J. Villadelprat [11] studied the period functions of centers of Hamiltonian potential systems and gave a criteria to bound the number of critical periods. Chicone [5] conjectured that the period function of the quadratic reversible centers have at most two critical periods. To illustrate the applicability of Theorem 1.1, we study the monotonicity of period functions of some quadratic reversible centers(see Section 3 below). In the literature there are a lot of papers dealing with the period functions of the quadratic centers satisfied some Picard-Fuchs differential equations (see [9,10,13,15,16,18,19] and references therein).
The paper is organized in the following way. In Section 2 we give the proof of Theorem 1.1 by using some results in [17]. In Section 3 we study the period functions of two quadratic reversible systems by applying Theorem 1.1.

The proof of main results
In what follows we give the following lemma in [17].
Lemma 2.1. [17] Under the assumptions (H1) and (H2), the following statements hold: The period of the periodic orbit γ h is given by In order to character the period function T (h) of the periodic orbit γ h we study the auxiliary function Lemma 2.2. The following statements hold: , then the period of the periodic orbit γ h is given by Moreover, Proof. It is clear that g(x) is well defined and analytic on (x I , x S ) and that On the other hand, we have Therefore g −1 (x) is well defined and analytic on (− √ h 0 , √ h 0 ) and g ′ (x) > 0 for all x ∈ (x I , x S ). Now turn to prove the statement (b). We make the variable z = g(x) in the expression of T (h) given by (2.1). Noting that for all h ∈ (0, h 0 ), it holds Making the variables z = √ h sin θ, θ ∈ (− π 2 , π 2 ), we get (2.2). Direct derivation with respect to h on (2.1), we have (2.3).
Since α(x) and G(x) G ′ (x) are analytic functions on (x I , x S ), we have that µ k (x) is an analytic function on (x I , x S ). The proof is finished.
This completes the proof of the result. The next result gives explicit expression of the derivative of the period function.

Lemma 2.4. Suppose that the function T (h) is the period function of the periodic orbit γ h of system (1.1). We take
and define ) .
Then, for any k ∈ N, the following equalities hold: Proof. We prove the result by induction on k. Making the variable This proves the case k = 1 in the statement. Suppose that the equality holds for k = n, it yields that where in the above equality we apply Lemma 2.3. This shows that the equality holds for k = n + 1. Therefore the proof is completed. In the following, we prove the main result of the paper. Proof of Theorem 1.1. By the Lemma 6, it follows that Taking the variables x = g −1 (τ ) in the second integral above, we have that ) .
, which implies that the period function of system (1.1) is monotone. This completes the proof of the result.
, then the number of the zeros S σ (µ k )(x) on (0, x S ) is equal to the number of the zeros , then the period function is a monotone function.

Applications
In the following examples, we shall apply Theorem 1 to prove the period functions of two reversible Lotka-Volterra systems Q LV 3 is monotone. (3.1) The first integral of (3.1) is H(x, y) = F (x)y 2 + G(x), where 2 and F (x) = 9 (3 + 4x) 2 .