Topological atlas of the Kovalevskaya top in a double field

TL;DR

This paper provides a comprehensive topological analysis of the Kovalevskaya top in a double field, introducing a topological atlas and classifying critical points and bifurcations in an integrable system with three degrees of freedom.

Contribution

It introduces the concept of a topological atlas for an irreducible integrable system and classifies all critical points and bifurcations for the Kovalevskaya top in a double field.

Findings

Complete classification of critical points and their types.

Construction of iso-energy invariants and bifurcation diagrams.

Determination of the number of critical periodic solutions.

Abstract

We fulfill the rough topological analysis of the problem of the motion of the Kovalevskaya top in a double field. This problem is described by a completely integrable system with three degrees of freedom not reducible to a family of systems with two degrees of freedom. The notion of a topological atlas of an irreducible system is introduced. The complete topological analysis of the critical subsystems with two degrees of freedom is given. We calculate the types of all critical points. We present the parametric classification of the equipped iso-energy diagrams of the complete momentum map pointing out all chambers, families of 3-tori, and 4-atoms of their bifurcations. Basing on the ideas of A.T. Fomenko, we introduce the notion of the simplified net iso-energy invariant. All such invariants are constructed. Using them, we establish, for all parametrically stable cases, the number of critical periodic solutions of all types and the loop molecules of all rank 1 singularities. The work was supported by the grants of the RFBR No. 13-01-97025 and 14-01-00119.