Small limit cycles bifurcating in pendulum systems under trigonometric perturbations

TL;DR

This paper investigates the bifurcation of small-amplitude limit cycles near the origin in perturbed pendulum systems with trigonometric polynomial perturbations, providing bounds on the number of such cycles.

Contribution

It establishes sharp upper bounds on the number of bifurcating limit cycles in perturbed pendulum systems with smooth and piecewise smooth polynomial perturbations.

Findings

Sharp upper bounds on the number of bifurcating limit cycles.

Analysis of both smooth and piecewise smooth perturbations.

Application of Melnikov function to determine cycle bifurcations.

Abstract

In this paper, we consider the bifurcation of small-amplitude limit cycles near the origin in perturbed pendulum systems of the form $\dot x= y$, $\dot y=-\sin(x)+\varepsilon Q(x,y)$, where $Q(x,y)$ is a smooth or piecewise smooth polynomial in the triple $(\sin(x),\cos(x), y)$ with free coefficients. We obtain the sharp upper bound on the number of positive zeros of its associated first order Melnikov function near $h=0$ for $Q(x,y)$ being smooth and piecewise smooth with the discontinuity at $y=0$, respectively.