Limit cycles in small perturbations of a planar piecewise linear Hamiltonian system with a non-regular separation line

Abstract

We study Poincaré bifurcation for a planar piecewise near-Hamiltonian system with two regions separated by a non-regular separation line, which is formed by two rays starting at the origin and such that the angle between them is α ∈ (0, π). The unperturbed system is a piecewise linear system having a periodic annulus between the origin and a homoclinic loop around the origin for all α ∈ (0, π). We give an estimation of the maximal number of the limit cycles which bifurcate from the periodic annulus mentioned above under n-th degree polynomial perturbations. Compared with the results in [13], where a planar piecewise linear Hamiltonian system with a straight separation line was perturbed by n-th degree polynomials, one more limit cycle is found. Moreover, based on our Lemma 2.5 we improve the upper bounds on the maximal number of zeros of the first order Melnikov functions derived in [19].