On the finite cyclicity of open period annuli

TL;DR

This paper proves that the maximum number of limit cycles bifurcating from an open period annulus in certain analytic vector fields is finite under multi-parameter deformations, extending understanding of cyclicity in dynamical systems.

Contribution

It establishes the finiteness of cyclicity for open period annuli in Hamiltonian or generic Darbouxian vector fields under analytic perturbations.

Findings

Finiteness of limit cycles bifurcating from open period annuli.

Applicable to Hamiltonian and Darbouxian vector fields.

Advances the cyclicity theory in real analytic dynamical systems.

Abstract

Let $\Pi$ be an open, relatively compact period annulus of real analytic vector field $X_0$ on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from $\Pi$ under a given multi-parameter analytic deformation $X_\lambda$ of $X_0$ is finite, provided that $X_0$ is either Hamiltonian, or generic Darbouxian vector field.