Alexander Gonchenko

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.

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 twodimensional 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.