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

TL;DR

This paper uses higher order Melnikov analysis to estimate the maximum number of limit cycles in planar piecewise linear systems with nonlinear switching curves, improving previous bounds for various cases.

Contribution

It introduces a higher order Melnikov approach for nonsmooth systems with nonlinear switching manifolds, providing new lower bounds for limit cycles.

Findings

Established new lower bounds for limit cycles: H(2)≥4, H(3)≥8, H(n)≥7 for even n≥4, H(n)≥9 for odd n≥5.

Extended Melnikov theory to nonsmooth systems with nonlinear switching curves.

Utilized Chebyshev systems to analyze limit cycle bifurcations.

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 $H(2)\geq 4,$ $H(3)\geq 8,$ $H(n)\geq7,$ for $n\geq 4$ even, and $H(n)\geq 9,$ for $n\geq 5$ odd. This improves all the previous results for $n\geq2.$ 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.