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<p>Using a higher-order averaging method, we rigorously prove the existence of periodic orbits bifurcating from both Hopf and zero-Hopf equilibria in the Coullet system. We also present numerical evidence showing that some of these periodic orbits may undergo a secondary period-doubling bifurcation, which sometimes initiates a period-doubling cascade that constitutes a classical route to chaotic behavior. Our analysis emphasizes the importance of examining secondary bifurcations of <i>k</i>-hyperbolic periodic solutions emerging from Hopf points.</p>

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A Note on K-Hyperbolic Periodic Solutions and Their Relation with Period-Doubling Bifurcations

  • 期刊论文

  • 发表日期：

    2026年5月13日

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  • Murilo R. Cândido,
  • Claudia Valls

摘要

Using a higher-order averaging method, we rigorously prove the existence of periodic orbits bifurcating from both Hopf and zero-Hopf equilibria in the Coullet system. We also present numerical evidence showing that some of these periodic orbits may undergo a secondary period-doubling bifurcation, which sometimes initiates a period-doubling cascade that constitutes a classical route to chaotic behavior. Our analysis emphasizes the importance of examining secondary bifurcations of k-hyperbolic periodic solutions emerging from Hopf points.