The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos

Abstract

We consider a special situation of the Hess-Appelrot case of the Euler-Poisson system which describes the dynamics of a rigid body about a fixed point. One has an equilibrium point of saddle type with coinciding stable and unstable invariant 2-dimensional separatrices. We show rigorously that, after a suitable perturbation of the Hess-Appelrot case, the separatrix connection is split such that only finite number of 1-dimensional homoclinic trajectories remain and that such situation leads to a chaotic dynamics with positive entropy and to the non-existence of any additional first integral.

On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point

“…The investigation of the Hess case of motion of a heavy rigid body with a fixed point can be reduced to the study of a vector field on a two-dimensional torus. In the series of papers [15][16][17][18], some topological properties of the phase flow of the problem are studied in the presence of perturbation (in the class of the Euler -Poisson system). In particular, limit cycles are considered near a critical circle case.…”