For the selected examples of the solutions rotate equations, the analysis of the static and dynamic equilibrium positions, as well as the structural stability of the phase portrait are phase portrait, vector rotators as a functions of the generalized angle coordinate of the deviator part of the mass moment vectors. The analogy between motions of heavy material point: 1) on the circle in vertical plane, 2) on the circle in vertical plane which rotate around vertical axis in the plane or out of the circle plane, and 3) on the sphere and corresponding motions case of the heavy rotor, as well as of the gyrorotor which rotates around two, or more axes with sections in one point or more points.
By using papers written by Ph. Holmes, as well as Smale-Birkhoff homoclinic theorem, and Hartman Grobman stable manifold theorem for fixed point about local stable and unstable manifold on the diffeomorfism with a hyperbolic homoclinic saddle fixed point the heavy forced rotor oscillatory motion in the neighborhood around hyperbolic points was studied. This forced motion under the action of the periodic couple excitations is stohastic like and chaotic like oscillatory process with sensitive dependence on the initial conditions. The Poincare maps are presented, as well as a Smale horse shoe maps.
It is studied nonlinear dynamics in the field of the turbulent damping for different gyrorotor system parameters. Equations of the phase trajectories family are determined, as well as special homoclinic orbits.