Melnikov analysis for planar piecewise linear vector fields with algebraic switching curve $y^n-x^m=0$

TL;DR

This paper investigates the maximum number of limit cycles in a planar piecewise linear system separated by an algebraic curve, providing lower bounds using Melnikov theory and Chebyshev systems.

Contribution

It offers the first lower estimates for the number of limit cycles in systems with algebraic switching curves of the form y^n - x^m=0.

Findings

Provides lower bounds for H(m,n) for all positive integers m,n.

Utilizes Melnikov theory and Chebyshev systems to analyze limit cycles.

Extends previous results to algebraic switching curves.

Abstract

This paper is devoted to the study of the maximum number of limit cycles, $H(m,n)$, of a planar piecewise linear differential system with two zones separated by the curve $y^n-x^m=0$, with $n,m$ being positive integers. More precisely, we provide a lower estimate of $ H(m,n)$. for all $m,n\in \mathbb{N}$, for piecewise linear perturbations of the linear center using some recent results about Chebyshev systems with positive accuracy and Melnikov Theory