On the number of limit cycles for generic Lotka-Volterra system and Bogdanov-Takens system under perturbations of piecewise smooth polynomials

TL;DR

This paper establishes upper bounds on the number of limit cycles bifurcating from specific Lotka-Volterra and Bogdanov-Takens systems under piecewise polynomial perturbations, using Picard-Fuchs equations to analyze Melnikov functions.

Contribution

It provides explicit bounds on limit cycles for these systems under polynomial perturbations, extending previous results to piecewise smooth cases.

Findings

Upper bounds of 36n-65 for n≥4 in Lotka-Volterra system

Upper bounds of 12n+6 for Bogdanov-Takens system

Explicit bounds for n=1,2,3 cases

Abstract

In this paper, we consider the bifurcation of limit cycles for generic L-V system ($\dot{x}=y+x^2-y^2\pm\frac{4}{\sqrt{3}}xy,~\dot{y}=-x+2xy$) and B-T system ($\dot{x}=y,~\dot{y}=-x+x^2$) under perturbations of piecewise smooth polynomials with degree $n$. Here the switching line is $y=0$. By using Picard-Fuchs equations, we bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles for generic L-V system and B-T system are respectively $36n-65~(n\geq4),~37,57,93~(n=1,2,3)$ and $12n+6$.