Andronov – Hopf and Andronov – Vitt – Shilnikov Bifurcations as a Route to Complex Dynamics in the Suslov Top

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.