Bifurcation of limit cycles by perturbing piecewise smooth integrable differential systems with four zones

TL;DR

This paper investigates how small perturbations can cause the emergence of limit cycles in four-zone piecewise smooth integrable systems, providing a method to estimate the maximum number of such bifurcations.

Contribution

It derives the first order Melnikov function for four-zone systems and applies it with Picard-Fuchs equations to bound bifurcated limit cycles.

Findings

Derived the first order Melnikov function for four-zone systems

Established an upper bound for bifurcated limit cycles

Applied the method to a specific system

Abstract

This paper deals with the problem of limit cycle bifurcations for piecewise smooth integrable differential systems with four zones. When the unperturbed system has a family of periodic orbits, the first order Melnikov function is derived which can be used to study the number of limit cycles bifurcated from the periodic orbits. As an application, using the first order Melnikov function and Picard-Fuchs equation, we obtain an upper bound of the number of bifurcated limit cycles of a concrete piecewise smooth differential system.