On the cyclicity of persistent hyperbolic polycycles

Abstract

In this work we consider families of smooth vector fields having a persistent polycycle with $n$ hyperbolic saddles. We derive the asymptotic expansion of the return map associated to the polycycle, determining explicitly its leading terms. As a consequence, explicit conditions on the leading terms allow us to determine the cyclicity of such polycycles. We then apply our results to study the cyclicity of a polycycle of a model with applications in Game Theory.