Chaotic dynamics of an unbalanced gyrostat

The free three-dimensional motion of an unbalanced gyrostat about the centre of mass is considered. The perturbed Hamiltonian for the case of small dynamical asymmetry of the rotor is written in Andoyer–Deprit canonical variables. The structure of the phase space of the unperturbed system is analysed, six forms of possible phase portraits are identified, and the equations of the phase trajectories are found analytically. Explicit analytical time dependences of the Andoyer–Deprit variables corresponding to heteroclinic orbits are obtained for all the phase portrait forms. The Melnikov function of the perturbed system is written for heteroclinic separatrix orbits using the analytical solutions obtained, and the presence of simple zeros is shown numerically. This provides evidence of intersections of the stable and unstable manifolds of the hyperbolic points and chaotization of the motion. Illustrations of chaotic modes of motion of the unbalanced gyrostat are presented using Poincaré sections.