On the number of limit cycles for polycycles $S^{(2)}$ and $S^{(3)}$ in quadratic Hamilton systems under perturbations of piecewise smooth polynomials

TL;DR

This paper investigates the maximum number of limit cycles bifurcating from quadratic Hamilton systems with specific structures under piecewise polynomial perturbations, providing explicit upper bounds based on polynomial degree.

Contribution

It introduces a novel approach using Picard-Fuchs equations and Chebyshev criteria to bound limit cycles in piecewise perturbed quadratic Hamilton systems.

Findings

Upper bounds of 25n+161 and 24n+126 limit cycles for systems S^{(2)} and S^{(3)}.

Bounded the zeros of the Melnikov function to control bifurcating limit cycles.

Applied advanced mathematical techniques to analyze bifurcations in piecewise smooth systems.

Abstract

In this paper, by using Picard-Fuchs equations and Chebyshev criterion, we study the bifurcate of limit cycles for quadratic Hamilton system $S^{(2)}$ and $S^{(3)}$: $\dot{x}= y+2axy+by^2$, $\dot{y}=-x+x^2-ay^2$ with $a\in(-\frac{1}{2},1)$, $b=(1-a)(1+2a)^{1/2}$ and $a=1$, $b=0$ respectively, under perturbations of piecewise smooth polynomials with degree $n$. The discontinuity is the line $y=0$. 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 $S^{(2)}$ and $S^{(3)}$ are respectively $25n+161$ $(n\geq3)$ and $24n+126$ $(n\geq3)$ (taking into account the multiplicity).