Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

This paper concerns limit cycle bifurcations by perturbing a piecewise linear Hamiltonian system. We first obtain all phase portraits of the unperturbed system having at least one family of periodic orbits. By using the first-order Melnikov function of the piecewise near-Hamiltonian system, we investigate the maximal number of limit cycles that bifurcate from a global center up to first order of 𝜀 .


Introduction and Main Results
Recently, piecewise smooth dynamical systems have been well concerned, especially in the scientific problems and engineering applications. For example, see the works of Filippov [1], Kunze [2], di Bernardo et al. [3], and the references therein. Because of the variety of the nonsmoothness, there can appear many complicated phenomena in piecewise smooth dynamical systems such as stability (see [4,5]), chaos (see [6]), and limit cycle bifurcation (see [7][8][9][10]). Here, we are more concerned with bifurcation of limit cycles in a perturbed piecewise linear Hamiltonian system: where > 0 is a sufficiently small real parameter, and = ( + , − , + , − ) ∈ ⊂ R 2( +1)( +2) with compact. Then system (1) has two subsystemṡ = + + ( , , ) , = + − ( , , ) , 2 Abstract and Applied Analysis which are called the right subsystem and the left subsystem, respectively. For = 0, systems (5a) and (5b) are Hamiltonian with the Hamiltonian functions, respectively, Note that the phase portrait of the linear systeṁ = , with 2 + 2 ̸ = 0 has possibly the following four different phase portraits on the plane (see Figure 1).
Then, one can find that system (1) | =0 can have 13 different phase portraits (see Figure 2) when at least one family of periodic orbits appears.
We remark that in Figure  It is easy to obtain the following Table 1 which shows conditions for each possible phase portrait appearing above. Also, cases (3), (5), (7), (9), and (13) in Figure 2 are equivalent to cases (2), (6), (8), (10), and (12), respectively, by making the transformation together with time rescaling = − . The authors Liu and Han [7] studied system (1) in a subcase of the case (1) of Figure 2 by taking 1 = 1 = 1, 0 = 0 = 0. By using the first order Melnikov function, they proved that the maximal number of limit cycles on Poincaré bifurcations is n up to first-order in . The authors Liang et al. [8] considered system (1) in the case (5) of Figure 2 by taking 1 = −1, 0 = 1, 1 = 1, and 0 = 0. By using the same method, they gave lower bounds of the maximal number of limit cycles in Hopf, and Homoclinic bifurcations, and derived an upper bound of the maximal number of limit cycles bifurcating from the periodic annulus between the center and the Homoclinic loop up to the first-order in . Clearly, the maximal number of limit cycles in the case (7) or (8) of Figure 2 is [( −1)/2] on Poincaré, Hopf and Homoclinic bifurcations up to first-order in , by using the first order Melnikov function.
This paper is organized as follows. In Section 2, we will provide some preliminary lemmas, which will be used to prove the main results. In Section 3, we present the proof of Theorem 1.

Coefficient conditions
(ii) (ℎ, ) in (11) can be expressed as Proof. We only prove (i) since (ii) can be verified in a similar way. By (11), we obtain which follows that by Green formula and (3) where Then, by Green formula again By (3), (4), and the above formulas, we have Combining (20)-(25) gives (13) and (14). Thus, the proof is ended.
Then, using Lemma 2 and (6) we can obtain the following three lemmas.