ON THE NUMBER OF LIMIT CYCLES BY PERTURBING A PIECEWISE SMOOTH HAMILTON SYSTEM WITH TWO STRAIGHT LINES OF SEPARATION∗

This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated.


Introduction and main results
Since the non-smooth phenomena exist widely in various practical applications and natural fields, such as automatic control, neural network, electrical engineering, economics, ecosystem, a big interest has appeared for studying bifurcation theory, especially bifurcation of limit cycles for planar piecewise smooth differential systems.
As pointed out by Kukucka [13], it is usually a nontrivial task to extend the bifurcation theory of smooth differential systems to non-smooth differential systems. So in recent years, the bifurcation of limit cycles for non-smooth differential systems with a straight line of separation has been investigated intensively and many innovative methods have been established. The Melnikov function method was extended to piecewise smooth differential system in [9,17]. In [17], Liu and Han derived the first order Melnikov function for planar piecewise smooth Hamilton systems which can be used to study the number of limit cycles for these systems. By using the Melnikov function method, Liang, Han and Romanovski [15] studied the number of limit cycles by perturbing a piecewise smooth linear Hamilton system with a generalized homoclinic loop around the origin, which takes the form ẋ = −y, y = 1 − x, x ≥ 0, ẋ = −y, y = x, x < 0. (1.1) For more results about this method, one can see [1,6,16,23,24,[26][27][28] and the references therein. Another important method called the averaging method which can be used to detect limit cycles for non-smooth differential systems is developed in [8,19,20]. More results on this topic can be found in [2,3,5,7,14,18,21]. Recently, Yang and Zhao [29] extended the Picard-Fuchs method to study the limit cycle bifurcations for piecewise smooth differential systems with a straight line of separation.
Under assumptions (I) and (II), {L h | h ∈ Σ} is a family of periodic orbits of system (1.6)| ε=0 and each L h is piecewise smooth. Without loss of generality, suppose that L h has an anticlockwise orientation, as shown in Fig. 1. From [22], the first order Melnikov function M (h) of system (1.6) has the following form (1.7) Further, we know from [11,22] that if M (h) has at most k zeros in h on the interval Σ, then (1.6) has at most k limit cycles bifurcated from the period annulus ∪ h∈Σ L h . In [12], by using the averaging method of first order, Itikawa et al. obtained the upper bounds of the number of limit cycles bifurcating from the periodic orbits of two kind of isochronous systems, when they are perturbed inside the discontinuous quadratic and cubic polynomials differential systems with two straight lines of separation, respectively. In [25], Xiong investigated the limit cycle bifurcation in perturbations of non-smooth Hamiltonian systems with 4 switching lines via multiple parameters.
In the present paper, motivated by the above references, we will study the number of limit cycles for a piecewise smooth Hamilton system with a generalized homoclinic loop under the perturbations of piecewise polynomials of degree n. More precisely, we consider the following piecewise smooth near-Hamilton system with two straight lines of separation A first integral of system (1.8)| ε=0 is with h ∈ (0, 1). When ε = 0, (1.8) has a family of piecewise smooth periodic orbits as follows with h ∈ (0, 1). If h → 1, L h approaches the origin which is an elementary center of parabolic-focus type, see [4,10]. If h → 0, L h approaches the generalized homoclinic loop L 0 with a saddle S(1, 0), see Fig. 2. Applying the above first order Melnikov function (1.7), we obtain the upper and lower bounds of the number of limit cycles which bifurcate from the period annulus of system (1.8)| ε=0 . Our main results are as follows: Theorem 1.1. The first order Melnikov function of system (1.8) is Theorem 1.2. Consider system (1.8) with |ε| small enough. Using the first order Melnikov function (1.7), the number of limit cycles bifurcating from the period annulus is no more than 2n + 5[ n−1 2 ] + 15, if n ≥ 2; 4 if n = 1, and at least 2n + 1 if n ≥ 1.
and g 2 (x, y) = g 3 (x, y), then (1.8) becomes (1.1) for ε = 0. The lower bound of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop is n + [ n+1 2 ], see Theorem 1.1 in [15]. The upper bound of the number of limit cycles also can be found in [15], but here we provide an alternative proof which is much more digestible.
The layout of the rest of this paper is as follows. The algebraic structure of the first order Melnikov function M (h) and the proof of Theorem 1.1 will be given in section 2. The Theorem 1.2 and Corollary 1.3 will be proved in sections 3 and 4.

Algebraic structure of M (h)
In the following, we will obtain the algebraic structure of M (h) of system (1.8).
Noting that (1.7) and we obtain that the first Melnikov function M (h) has the form For h ∈ (0, 1) and i, j ∈ N, we denote It is easy to check that Proof. For the sake of clearness, we split the proof into two steps.
(1) We first assert that where τ i,j and σ i,j are arbitrary real constants, ϕ n+1 (u) is a polynomial in u of degree n + 1 and ψ n+1 (u) is a polynomial in u of degree n + 1 without constant term.
In fact, Let Ω be the interior of Fig. 1. Using the Green's Formula, we have for i ≥ 0 and j ≥ 1 In a similar way, we have for i ≥ 0 and j ≥ 1 On the other hand, we have for i ≥ 0 and j = 0 (2.7) From (2.1), (2.2) and (2.5)-(2.7), we obtain where Thus, τ i,j and σ i,j can be chosen arbitrarily. By direct computation, we have where ϕ n+1 (u) is a polynomial of u with degree n + 1 and ψ n+1 (u) is a polynomial of u with degree n + 1 without constant term. Substituting (2.9) into (2.8) gives (2.4).
Proof of the Theorem 1.1. By some straightforward calculations, we have
If n is an odd number, we can prove that system (1.8) has 2n + 1 limit cycles for 0 < h < 1 in a similar way. This ends the proof of Theorem 1.2.