Bifurcation of limit cycles from a quadratic global center with two switching lines

TL;DR

This paper extends the Picard-Fuchs method to analyze how two switching lines influence the bifurcation of limit cycles in quadratic systems, providing a simplified approach and bounds on their number.

Contribution

It introduces a generalized Picard-Fuchs equation method for systems with two switching lines, simplifying computations and revealing the impact of switching lines on limit cycle bifurcations.

Findings

Derived explicit Melnikov functions for perturbed systems

Established upper bounds on the number of bifurcating limit cycles

Showed the influence of switching lines on bifurcation behavior

Abstract

In this paper, we generalize the Picard-Fuchs equation method to study the bifurcation of limit cycles of perturbed differential systems with two switching lines. We obtain the detailed expression of the corresponding first order Melnikov function which can be used to get the upper bound of the number of limit cycles for the perturbed system by using Picard-Fuchs equation. It is worth noting that we greatly simplify the computations and this method can be applied to study the number of limit cycles of other differential systems with two switching lines. Our results also show that the number of switching lines has essentially impact on the number of limit cycles bifurcating from a period annulus.