The Hess–Appelrot system. II. Perturbation and limit cycles

We study the Hess–Appelrot case of the Euler–Poisson system which describes dynamics of a rigid body around a fixed point. It is well known that in this case there is an invariant surface S in the phase space. In our previous paper (Lubowiecki and Żołądek, 2011 [6]) we proved that this surface is a torus and restricted to it dynamics is either hyperbolic or parabolic or elliptic quasi-periodic or elliptic periodic. We proved that the invariant torus is normally hyperbolic when the torus is close to a ‘critical circle’ and the motion is 1:q resonant. In this paper we consider perturbation of the Hess–Appelrot system (within the Euler–Poisson class) near the above situation. We prove existence of an invariant surface close to S and we study limit cycles on the perturbed surface. We estimate the number of such cycles by analysis of some non-standard Melnikov integrals.