The number of limit cycles of a kind of piecewise quadratic systems with switching curve y = xm

Abstract

We first study the expressions for the second-order Melnikov functions for planar piecewise smooth near-Hamiltonian systems with multiple zones separated by multiple switching curves through the origin. Next, up to the second-order Melnikov functions, we consider the number of limit cycles Z(m,2) of the system x˙=y,y˙=−x with two zones separated by the curve y=xm(2≤m∈N) under the piecewise perturbations of polynomials in x and y with degree 2. It is proved that Z(2,2)=9, 13≤Z(4,2)≤19, and Z(2k,2)≥14 for k≥3 and that Z(2k+1,2)=15 for k≥2. The results are new and improve some results in the literature.