The period of the limit cycle bifurcating from a persistent polycycle

TL;DR

This paper analyzes the bifurcation of a limit cycle from a persistent polycycle in planar polynomial vector fields, showing exponential approach and infinite period growth near critical parameter values, with behavior depending on generic conditions.

Contribution

It provides a detailed description of the bifurcation behavior of limit cycles from polycycles, including exponential convergence and period divergence, extending known cyclicity results.

Findings

Limit cycle approaches the polycycle exponentially fast.

Period of the limit cycle tends to infinity as 1/|r(μ)-1| near bifurcation.

Behavior of the period depends on generic conditions.

Abstract

We consider smooth families of planar polynomial vector fields $\{X_\mu\}_{\mu\in\Lambda}$, where $\Lambda$ is an open subset of $\mathbb{R}^N$, for which there is a hyperbolic polycycle $\Gamma$ that is persistent (i.e., such that none of the separatrix connections is broken along the family). It is well known that in this case the cyclicity of $\Gamma$ at $\mu_0$ is zero unless its graphic number $r(\mu_0)$ is equal to one. It is also well known that if $r(\mu_0)=1$ (and some generic conditions on the return map are verified) then the cyclicity of $\Gamma$ at $\mu_0$ is one, i.e., exactly one limit cycle bifurcates from $\Gamma$. In this paper we prove that this limit cycle approaches $\Gamma$ exponentially fast and that its period goes to infinity as $1/|r(\mu)-1|$ when $\mu\to\mu_0.$ Moreover, we prove that if those generic conditions are not satisfied, although the cyclicity may be exactly 1, the behavior of the period of the limit cycle is not determined.