The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity
Paweł Lubowiecki Henryk Żołądek
Journal of Geometric Mechanics 2012, 4(4): 443-467 doi: 10.3934/jgm.2012.4.443
We study the Hess--Appelrot case of the Euler--Poisson system which describes dynamics of a rigid body about a fixed point. We prove existence of an invariant torus which supports hyperbolic or parabolic or elliptic periodic or elliptic quasi--periodic dynamics. In the elliptic cases we study the question of normal hyperbolicity of the invariant torus in the case when the torus is close to a `critical circle'. It turns out that the normal hyperbolicity takes place only in the case of $1:q$ resonance. In the sequent paper [16] we study limit cycles which appear after perturbation of the above situation.
keywords: invariant torus Hess-Appelrot system normal hyperbolicity.
The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos
Radosław Kurek Paweł Lubowiecki Henryk Żołądek
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 1955-1981 doi: 10.3934/dcds.2018079

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.

keywords: Hess-Appelrot system splitting of separatrices chaotic dynamics

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