Efrosiniia Karatetskaia

We study the chaotic dynamics of the Suslov top, a classical nonholonomic system. This system possesses a rich bifurcation structure, which we explore numerically via Lyapunov diagrams and bifurcation analysis in a two-parameter plane. This structure is organized by a codimension-2 point where an equilibrium state has eigenvalues $(0, i\omega,−i\omega)$ and a homoclinic loop. From this point, two important bifurcation curves originate: the Andronov – Hopf bifurcation curve, corresponding to the birth of a stable periodic orbit from an equilibrium state, and the Andronov – Vitt – Shilnikov bifurcation curve, corresponding to the birth of a stable periodic orbit from a homoclinic loop of a saddle-node equilibrium. We show how chaotic attractors and repellers emerge from equilibrium states, grow, and merge into mixed dynamics. We also discuss the connection of this organizing codimension-2 point with the blue sky catastrophe phenomenon: the birth of a stable periodic orbit of infinite period and length.

We study hyperchaotic attractors characterized by three positive Lyapunov exponents in numerical experiments. In order to possess this property, periodic orbits belonging to the attractor should have a three-dimensional unstable invariant manifold. Starting with a stable fixed point we describe several bifurcation scenarios that create such periodic orbits inside the attractor. These scenarios include cascades of alternating period-doubling and Neimark – Sacker bifurcations which, as we show, naturally appear near the cascade of codimension-2 period-doubling bifurcations, when periodic orbits along the cascade have multipliers $(-1, e^{i \phi}, e^{-i \phi})$. The proposed scenarios are illustrated by examples of the threedimensional Kaneko endomorphism and a four-dimensional Hénon map.

We provide a detailed bifurcation analysis in a three-dimensional system describing interaction between tumor cells, healthy tissue cells, and cells of the immune system. As is well known from previous studies, the most interesting dynamical regimes in this model are associated with the spiral chaos arising due to the Shilnikov homoclinic loop to a saddle-focus equilibrium [1–3]. We explain how this equilibrium appears and how it gives rise to Shilnikov attractors. The main part of this work is devoted to the study of codimension-two bifurcations which, as we show, are the organizing centers in the system. In particular, we describe bifurcation unfoldings for an equilibrium state when: (1) it has a pair of zero eigenvalues (Bogdanov – Takens bifurcation) and (2) zero and a pair of purely imaginary eigenvalues (zero-Hopf bifurcation). It is shown how these bifurcations are related to the emergence of the observed chaotic attractors.