On the number of limit cycles for perturbed pendulum equations

TL;DR

This paper investigates the maximum number of limit cycles that can bifurcate from periodic orbits in perturbed pendulum equations, providing bounds based on polynomial degrees and elliptic functions.

Contribution

It introduces new upper bounds on the number of limit cycles for perturbed pendulum equations using Abelian integrals and Chebyshev systems analysis.

Findings

Derived upper bounds for zeros of Melnikov functions in oscillatory and rotary regions.

Expressed Abelian integrals in terms of polynomials and elliptic functions.

Identified subfamilies forming Chebyshev systems to obtain sharp bounds.

Abstract

We consider perturbed pendulum-like equations on the cylinder of the form $ \ddot x+\sin(x)= \varepsilon \sum_{s=0}^{m}{Q_{n,s} (x)\, \dot x^{s}}$ where $Q_{n,s}$ are trigonometric polynomials of degree $n$, and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case $\varepsilon=0$ in terms of $m$ and $n$. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. Some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be Chebyshev systems.