In this paper a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom. The methods developed are illustrated by examples of several integrable problems related to the classical Euler-Poisson equa- tions, the motion of a rigid body in a fluid, and the dynamics of gaseous expanding ellipsoids. These topological methods also enable one to find non-degenerate periodic solutions of integrable systems, which is especially topical in those cases where no general solution (for example, by separation of variables) is known. Bibliography: 82 titles.

https://iopscience.iop.org/article/10.1070/RM2010v065n02ABEH004672/pdf

Topology and stability of integrable systems

The dynamical motion of a gyrostat for the irrational frequency case

Топология и устойчивость интегрируемых систем@@@Topology and stability of integrable systems

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting.

A.P. Markeyev

Papers

The pendulum-like motions of a rigid body in the Goryachev-Chaplygin case