Bifurcation of limit cycles from a cubic reversible isochrone

TL;DR

The paper addresses limit-cycle bifurcation from a uniformly isochronous center of a cubic reversible system under polynomial perturbations. It introduces the Abelian integral $I(h)$ along the symmetric period annulus and derives its algebraic structure by expressing it through recurrence relations and ultimately via complete elliptic integrals $K(k)$ and $E(k)$. Using the Argument Principle and Picard-Fuchs-type relations, it establishes an explicit upper bound of $22n+6$ limit cycles bifurcating from the period annulus, and shows that for small $n$ configurations can yield $1$ or $2$ limit cycles. Numerical simulations are provided to confirm the existence of limit cycles and to illustrate multicycle scenarios for particular coefficient choices. The work clarifies the scope of Abelian-integral methods for cubic reversible systems and outlines open questions when extending to more general polynomial perturbations.

Abstract

This paper is devoted to study the limit cycle problem of a cubic reversible system with an isochronous center, when it is perturbed inside a class of polynomials. An upper bound of the number of limit cycles is obtained using the Abelian integral. The algebraic structure of the Abelian integral is acquired thanks to some iterative formulas, which differs in many aspects from other methods. Some numerical simulations verify the existence of limit cycles.