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

Abstract

We study bifurcation scenarios leading to the so-called Shilnikov singular attractors as 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 invariant curve as a result of a supercritical Neimark – Sacker bifurcation, and the hard one relates to the immediately occurring big invariant curve after disappearance of a stable fixed point at the saddle-node bifurcation. We study both these mechanisms and trace subsequent scenarios of breakdown of the invariant curve and chaos development leading to the emergence of Shilnikov singular attractors. Additionally, we study geometrical peculiarities of these attractors, such as structures of rotating patterns of orbits inside the Shilnikov singular funnel and present a two-parametric analysis with Lyapunov exponents and minimal a distance between the chaotic attractor and the unstable focus.

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