Topology and stability of integrable systems. (English. Russian original) Zbl 1202.37077

Bolsinov, A. V.; Borisov, A. V.; Mamaev, I. S.

Russ. Math. Surv. 65, No. 2, 259-317 (2010); translation from Usp. Mat. Nauk 65, No. 2, 71-132 (2010).

The present paper is devoted to a general approach to the stability problem for integrable dynamical systems, based on the systematic use of methods from topological analysis, especially for the case of two degrees of freedom. To illustrate their method, the authors constructed a new tool called the bifurcation complex for two classical problems in the dynamics of a rigid body, namely the Goryachev-Chaplygin top and the Clebsch system (which is an integrable case of the motion of a rigid body in a fluid), as well as for the Gaffet system describing the dynamics of a gaseous ellipsoid filled with a monatomic ideal gas.

Reviewer: Mircea Crâşmăreanu (Iaşi)

MSC:

37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37-02	Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37J05	Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37J20	Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37J25	Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
70E40	Integrable cases of motion in rigid body dynamics
70E50	Stability problems in rigid body dynamics
70G40	Topological and differential topological methods for problems in mechanics
70H14	Stability problems for problems in Hamiltonian and Lagrangian mechanics

Keywords:

topology; stability; periodic trajectory; critical set; bifurcation set; bifurcation diagram