Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear switching curve

Abstract

In this paper, we are interested in providing lower estimations for the maximum number of limit cycles $H(n)$ that planar piecewise linear differential systems with two zones separated by the curve $y=x^n$ can have, where $n$ is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that $H(2)\ge 4,$ $H(3)\ge 8,$ $H(n)\ge 7,$ for $n\ge 4$ even, and $H(n)\ge 9,$ for $n\ge 5$ odd. This improves all the previous results for $n\ge 2.$ Our analysis is mainly based on some recent results about Chebyshev systems with positive accuracy and Melnikov theory, which will be developed at any order for a class of nonsmooth differential systems with nonlinear switching manifold.