On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields

TL;DR

This paper proves that in a family of planar vector fields, a heteroclinic saddle loop can generate only finitely many limit cycles through perturbations, establishing a bound on their number.

Contribution

It establishes a finite upper bound on the number of limit cycles arising from perturbations of two-saddle cycles in analytic planar vector fields.

Findings

Finite number of limit cycles generated by saddle loops

Bound on limit cycles depends on perturbation parameters

Analyticity ensures finiteness of bifurcations

Abstract

We prove that every heteroclinic saddle loop (a two-saddle cycle) occurring in an analytic finite-parameter family of plane analytic vector fields, may generate no more than a finite number of limit cycles within the family.