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<p>We study bifurcation scenarios leading to the so-called Shilnikov singular attractorsas a result of breakdown of closed invariant curves in the Chialvo map, which is a two-dimensional endomorphism demonstrating neuron-like dynamics. We show that two different routes, soft and hard, of the emergence of such closed invariant curves can be traced here: the soft one corresponds to the birth of an invariantcurve as a result of a supercritical Neimark – Sacker bifurcation, and the hard one relatesto the immediately occurring big invariant curve after disappearance of a stable fixed pointat the saddle-node bifurcation. We study both these mechanisms and trace subsequent scenariosof breakdown of the invariant curve and chaos development leading to the emergence ofShilnikov singular attractors. Additionally, we study geometrical peculiarities ofthese attractors, such as structures of rotating patterns of orbits inside the Shilnikovsingular funnel and present a two-parametric analysis with Lyapunov exponents and minimala distance between the chaotic attractor and the unstable focus.</p>

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Destruction of Invariant Curves and Singular Shilnikov Attractors in the Chialvo Map

  • 期刊论文

  • 发表日期：

    2025年12月30日

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  • Nataliya V. Stankevich,
  • Alexander S. Gonchenko,
  • Elena S. Popova,
  • Elmira R. Bagautdinova

摘要

We study bifurcation scenarios leading to the so-called Shilnikov singular attractorsas a result of breakdown of closed invariant curves in the Chialvo map, which is a two-dimensional endomorphism demonstrating neuron-like dynamics. We show that two different routes, soft and hard, of the emergence of such closed invariant curves can be traced here: the soft one corresponds to the birth of an invariantcurve as a result of a supercritical Neimark – Sacker bifurcation, and the hard one relatesto the immediately occurring big invariant curve after disappearance of a stable fixed pointat the saddle-node bifurcation. We study both these mechanisms and trace subsequent scenariosof breakdown of the invariant curve and chaos development leading to the emergence ofShilnikov singular attractors. Additionally, we study geometrical peculiarities ofthese attractors, such as structures of rotating patterns of orbits inside the Shilnikovsingular funnel and present a two-parametric analysis with Lyapunov exponents and minimala distance between the chaotic attractor and the unstable focus.