LIMIT CYCLE BIFURCATIONS IN DISCONTINUOUS PLANAR SYSTEMS WITH MULTIPLE LINES∗

Abstract In this paper, the limit cycle bifurcation problem is investigated for a class of planar discontinuous perturbed systems with n parallel switch lines. Under the assumption that the unperturbed system has a family of periodic orbits crossing all of the lines, an explicit expression of the first order Melnikov function along the periodic orbits is presented, which plays an important role in studying the problem of limit cycle bifurcations. As an application of the established method, the maximal number of limit cycles of a discontinuous system is considered.


Introduction
Consider a piecewise smooth system of the froṁ
Definition 2.1. If a planar system has n subsystems and each subsystem is a near-Hamiltonian (resp. Hamiltonian) system on the plane, then we call this system an n-piecewise near-Hamiltonian (resp. Hamiltonian) system.
From Definition 2.1, one can know that system (2.2) (resp. (2.3)) is an (n + 1)piecewise near-Hamiltonian (resp. Hamiltonian) system. And system (2.2)| ε=0 or (2.3) has a piecewise Hamiltonian function defined by H(x, y), i.e. (2.6) Regarding system (2.3), we make the following three assumptions: 3) has a period annulus A consisting of a one-parameter family of clockwise periodic orbits For h ∈ J , each Γ h crosses the straight line l i : x = x i two times clockwise, having the intersection points denoted by Figure 1.
where i = 1, 2, · · · , n − 1. From the above assumptions, one can see that the (x, y)-plane has been split into n + 1 subregions by the straight lines l 1 , l 2 , · · · , l n . That is to say, On each subregion I i ×R, it defines a C ∞ near-Hamiltonian system. And there exists a family of periodic Γ h , h ∈ J passing through each subregion with a clockwise orientation. Now, we investigate the unperturbed system (2.2) under the conditions (A1)-(A3). By continuous dependency of discontinuous planar systems on initial data established in [2], the positive orbit of system (2.2) starting from A 1 must intersect the straight lines l 1 , · · · , l n successively with points B 1ε (h) = (x 1 , b 1ε (h)), · · · , B nε (h) = (x n , b nε (h)), then it turns and crosses the lines l n , · · · , l 1 again with points A nε (h) = (x n , a nε (h)), · · · , A 1ε (h) = (x 1 , a 1ε (h)) respectively, see Figure 2. Obviously, in view of (A3), one knows that It is not hard to see that from the point A 1 (h) to the point A 1ε (h) on the straight line l 1 , it results in a return map or Poincaré map, denoted by P(h, ε), Based on these, we can define a function below On account of the definition of H 0 (x, y), together with (2.7), one can find that the On the other hand, by the differential mean value theorem, Recall that, similar to the case of continuous systems, a limit cycle is an isolated periodic orbit of the discontinuous systems. Therefore, for ε > 0 small, system (2. It is easy to prove that, if the first non-zero M i (h) has a zero of odd multiplicity in h, then for ε > 0 small enough, F (h, ε) also has a zero having an odd multiplicity near h. This means that one can investigate the number of isolated zeros of the first non-zero M i (h) to obtain the number of limit cycles emerging from the period annulus A. Recently, the authors of [17] have derived the explicit expression of M 1 (h) for a 2-piecewise near-Hamiltonian system and gave the corresponding applications.
This paper focuses on presenting the general expression of M 1 (h) for an (n + 1)piecewise near-Hamiltonian system, n ≥ 1, n ∈ N. As an application, we study the limit cycle bifurcation for a 3-piecewise near-Hamiltonian system.
Then, using the same idea of [17] or by Theorem 2.2 in [16], it is easy to obtain the following lemma.
Particularly, we have for n = 1, and for n = 2, we have having an odd multiplicity, then for ε > 0 small, system (2.2) has at least one limit cycle near Γ h0 .
Usually, the boundary of the period annulus A is a center or a polycycle. For example, the following 2-piecewise Hamiltonian system The boundary of A is a generalized homoclinic loop (as h → 1 2 ) or the origin (as h → 0). Here, the origin is an elementary center, see Definition 1.4 of [13]. Similar to smooth systems [14,15,32,33], one can investigate the asymptotic expansion of M (h) near the boundary to study the homoclinic loop bifurcation or the Hopf bifurcation, see [18,19,29]. Now, we apply Lemma 2.1 to study the limit cycle bifurcation of a 3-piecewise near-Hamiltonian system. For definiteness, take Φ(x, y) in (2.1) as follows (2.11) where f (x) and g(x) are given in (2.11), i.e.
In view of Lemma 2.1, one can obtain that Theorem 2.1. For ε > 0 small, system (2.12) can have 7m + 3 limit cycles.
From Theorem 2.1, we conclude that a 3-piecewise linear near-Hamiltonian system can have 3 limit cycles. While, for a 2-piecewise linear near-Hamiltonian, we only find 2 limit cycles, see [7,29]. This implies that one can find more limit cycles by the Melnikov function method by adding a switch line. The proof of Theorem 2.1 will be given in the next section.

Proof of Theorem 2.1
Clearly, system (2.12) has the following three subsystemṡ It is easy to see that system (2.12)| ε=0 has three families of periodic orbits given by
Then, associated to the three families of periodic orbits, one has three the first order Melnikov functions as follows and h ∈ (0, +∞).
For |h| > 0 small, , r ≥ 2, (3.25) (3.22) can be expanded as where Similarly, M * (h) can be expressed as for 0 < h 1 where C i and D i are the same as in (3.27).