Multiplicity of limit cycles that appear after perturbations of hyperbolic polycycles

TL;DR

This paper investigates how many limit cycles can emerge from hyperbolic polycycles under perturbations, establishing an upper bound related to the polycycle's separatrix connections in generic finite-parameter families.

Contribution

It provides a new upper bound on the multiplicity of limit cycles emerging from hyperbolic polycycles after perturbation, linking it to the number of separatrix connections.

Findings

Limit cycles' multiplicity is bounded by the number of separatrix connections.

The result applies to generic finite-parameter families of perturbations.

The paper advances understanding of bifurcations in dynamical systems with polycycles.

Abstract

We consider the multiplicity of limit cycles that appear when a hyperbolic polycycle is perturbed. We prove, in particular, that if such unfolding happens in generic finite-parameter families, the multiplicity of every new limit cycle does not exceed the number of separatrix connections in the polycycle.