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<p>In this work we consider families of smooth vector fields having a persistent polycycle with <i>n</i> hyperbolic saddles. We derive the asymptotic expansion of the return map associated to the polycycle, determining its leading terms. As a consequence, we obtain conditions on the leading terms that allow us to determine the cyclicity of such polycycles. A general sufficient condition for the stability of polycycles with graphic number equal to one is also obtained. We then apply our results to study the cyclicity of a polycycle of a model with applications in Game Theory.</p>

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On the cyclicity of persistent hyperbolic polycycles

  • 期刊论文

  • 发表日期：

    2025年12月12日

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  • Lucas Queiroz Arakaki,
  • Paulo Santana

摘要

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 its leading terms. As a consequence, we obtain conditions on the leading terms that allow us to determine the cyclicity of such polycycles. A general sufficient condition for the stability of polycycles with graphic number equal to one is also obtained. We then apply our results to study the cyclicity of a polycycle of a model with applications in Game Theory.