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JGM

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.

DCDS

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.

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