FURTHER STUDIES ON LIMIT CYCLE BIFURCATIONS FOR PIECEWISE SMOOTH NEAR-HAMILTONIAN SYSTEMS WITH MULTIPLE PARAMETERS∗

This paper investigates the limit cycle bifurcations for piecewise smooth near-Hamiltonian systems with multiple parameters. The formulas for the second and third term in expansions of the first order Melnikov function are derived respectively. The main results improve some known conclusions.


Introduction and Main Results
Piecewise smooth systems are frequently encountered in practical applications, such as control systems and engineering [1,12,21]. In recent years, there are lots of works on studying the number of limit cycles and their relative positions of nonsmooth dynamical systems on the plane and have obtained many meaningful results [4,5,10]. It is well known that the Melnikov method is a useful tool to determine the number of limit cycles bifurcating from a family of periodic orbits of the unperturbed systems. The authors in [16] established a formula for the first order Melnikov function for planar piecewise smooth systems, which plays an important role in estimating the number of limit cycles, see for instance [13,24]. For high-dimensional piecewise smooth near-integrable systems, the authors of [20] established the Melnikov function theory and gave an expression for the first order Melnikov vector function. We note that the averaging method developed in [7,14,15,18] is another common technique. For some applications of this method see [2,17,19] for example. It was proved in [8] that the averaging method is equivalent to the Melnikov function method for studying the number of limit cycles of planar analytic (or C ∞ ) near-Hamiltonian systems.
In this paper, we consider a piecewise smooth near-Hamiltonian system with multiple parameters of the following form: H + y (x, y, λ) + εp + (x, y, λ) −H + x (x, y, λ) + εq + (x, y, λ) , x > 0, where H ± , p ± and q ± are C ∞ functions, λ and ε are both sufficiently small real parameters with 0 < ε ≪ λ ≪ 1. Suppose system (1.1) satisfies the following assumptions as in [9,16,20]: (I) There exist an interval J = (α, β) and two points A λ (h) = (0, a(h, λ)) and defines an orbital arc L − h starting from B λ (h) and ending at A λ (h), such that system (1.1)| ε=0 has a family clockwise oriented periodic orbits Under the conditions (I)-(III), we have the first order Melnikov function of system (1.1) from [9,16] (1.2) Sometimes the system we consider has the following form where the functions f ± 1 , f ± 2 , g ± 1 and g ± 2 are C ∞ functions such that the above unperturbed system has integrating factors µ 1 and µ 2 and first integrals H + and H − respectively for x ≥ 0 and x < 0, satisfying Then the above differential equation is equivalent to a near-Hamiltonian system of the form (1.1), and the corresponding Melnikov function has the form In system (1.1), the functions H ± , p ± and q ± depend on another small parameter λ leading to the dependence of the function M on λ. Then for λ > 0 small (1. 3) The function M (h, λ) can be used to study not only Poincaré bifurcation (bifurcation of limit cycles from a period annulus) but also Hopf bifurcation and homoclinic and heteroclinic bifurcations. The formulas of M 1 (h) and M 2 (h) were obtained in [11] for smooth case. If H − (A λ (h), λ) = h, the author [22] gave the formulas of M 1 (h) and M 2 (h), which has some applications, see for example [3,23]. Our main task in this paper is to remove the condition H − (A λ (h), λ) = h and give expressions of M 1 (h) and M 2 (h) under the conditions (I)-(III). For the purpose, assume the functions H ± , p ± and q ± have the following form for λ > 0 small Then from (1.2), (1.3) and above expansions, it is easy to see that For convenience, we introduce some notations below. Denote (1.6) The main results are as follows.

Theorem 1.2. Under the conditions (I)-(III), suppose further that there exist a region U and C
We have In the next two sections, we provide proofs of the above theorems and present an example showing an application of our main results, respectively.

Proof of main results
Enlightened by the idea in [22], we first present a preliminary lemma, which will be used in deducing the expressions of M 1 (h) and M 2 (h).
wherep ± andq ± are C ∞ functions in (x, y) and independent of λ. Then (2.1) Proof. The first formula in (2.1) was obtained in [22], thus we omit its proof here and only prove the second one in (2.1). By using Green formula twice, we havê For the sake of simplicity, assume that B λ A λ can be represented as y = y − 1 (x, h, λ) and A direct computation shows that Obviously, On the other hand, noting that This completes the proof. and where for i = 0, 1, 2.
In order to deduce the formula for M 2 (h), we first prove the following helpful lemmas.

Lemma 2.2. Assume
wherep ± andp ± are given in Lemma 2.1. Further suppose that there exist a region U and C ∞ functionsp ± (x, y) andq ± (x, y) defined on U such that (2.12) Then Proof. The first formula in (2.13) can be found in [22], thus we just prove the second one. From (1.4) and (2.12), I − (h, λ) can be written in an equivalent way as Thus, we obtain (2.14) By (2.12) and using Lemma 2.1 we have Taking λ = 0 in (2.15) and substituting the result into (2.14) give the second formula of (2.13). This ends the proof.

Lemma 2.3. Let
where g is given in (1.6). Then Proof. By the definition of the orbit B λ A λ , we can rewrite J(h, λ) as

From (2.5) we can obtain
Hence, by (2.18) together with some calculations, we obtain from (2.17) Using Lemma 2.2 and (2.11), one can see