On the number of limit cycles bifurcating from the linear center with an algebraic switching curve

TL;DR

This paper investigates how the degree of algebraic switching curves influences the maximum number of limit cycles bifurcating from a linear center in piecewise linear systems, providing bounds based on polynomial perturbations.

Contribution

It introduces bounds on the number of bifurcating limit cycles considering the degree of algebraic switching curves, a novel aspect in piecewise linear system analysis.

Findings

Degree of switching curve affects limit cycle count

Established upper and lower bounds for bifurcating limit cycles

Analyzed the impact of polynomial perturbations of degree n

Abstract

This paper studies the family of piecewise linear differential systems in the plane with two pieces separated by a switching curve $y=x^{m}$, where $m>1$ is an arbitrary positive. By analysing the first order Melnikov function, we give an upper bound and an lower bound of the maximum number of limit cycles which bifurcate from the period annulus around the origin under polynomial perturbations of degree $n$. The results shows that the degree of switching curves affect the number of limit cycles.