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

TL;DR

This paper investigates the maximum number of limit cycles that can bifurcate from a Bogdanov-Takens system when subjected to perturbations by piecewise smooth polynomials, providing explicit upper bounds based on polynomial degree and switching curves.

Contribution

It establishes new upper bounds for the number of bifurcating limit cycles in the Bogdanov-Takens system under piecewise polynomial perturbations, considering various switching curves and lines.

Findings

Upper bounds for limit cycles with switching curve $x=y^{2m}$ are $(39m+36)n+77m+21(m extgreater=2)$ and $50n+52$ for $m=1$.

Maximum of 11 limit cycles with switching lines $x=0$ and $y=0$, which is attainable.

Bounds depend on polynomial degree and switching curve, advancing understanding of bifurcation behavior.

Abstract

In this paper, we study the bifurcate of limit cycles for Bogdanov-Takens system($\dot{x}=y$, $\dot{y}=-x+x^{2}$) under perturbations of piecewise smooth polynomials of degree $2$ and $n$ respectively. 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 with switching curve $x=y^{2m}$($m$ is a positive integral) are $(39m+36)n+77m+21(m\geq 2)$ and $50n+52(m=1)$ (taking into account the multiplicity). The upper bounds number of limit cycles with switching lines $x=0$ and $y=0$ are 11 (taking into account the multiplicity) and it can be reached.