Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

Maoan Han,Jiangbin Chen

doi:10.1007/s12346-022-00567-0

Abstract

In this paper, we study limit cycle bifurcations for planar piecewise smooth near-Hamiltonian systems with nth-order polynomial perturbation. The piecewise smooth linear differential systems with two centers formed in two ways, one is that a center-fold point at the origin, the other is a center-fold at the origin and another unique center point exists. We first explore the expression of the first order Melnikov function. Then by using the Melnikov function method, we give estimations of the number of limit cycles bifurcating from the period annulus. For the latter case, the simultaneous occurrence of limit cycles near both sides of the homoclinic loop is partially addressed.

Highlights

Introduction and Main ResultsRecently, piecewise smooth dynamical systems have been well concerned, especially in the scientific problems and engineering applications
By using the first order Melnikov function, they proved that the maximal number of limit cycles on Poincarebifurcations is n up to first-order in ε
This paper focuses on studying the limit cycle bifurcations of system (1) in the case (1) of Figure 2 by using the first order Melnikov function

Summary

Introduction and Main Results

Piecewise smooth dynamical systems have been well concerned, especially in the scientific problems and engineering applications. Ẏ = −ax − b, with a2 +b2 ≠ 0 has possibly the following four different phase portraits on the plane (see Figure 1). By using the first order Melnikov function, they proved that the maximal number of limit cycles on Poincarebifurcations is n up to first-order in ε. The maximal number of limit cycles in the case (7) or (8) of Figure 2 is [(n−1)/2] on Poincare, Hopf and Homoclinic bifurcations up to first-order in ε, by using the first order Melnikov function. This paper focuses on studying the limit cycle bifurcations of system (1) in the case (1) of Figure 2 by using the first order Melnikov function.

Preliminary Lemmas

Proof of Theorem 1