Bifurcations of a nonlinear spherical pendulum with vibrating suspension point

Abstract

This paper considers a nonlinear spherical pendulum whose suspension point performs high-frequency spatial vibrations. The dynamics of this pendulum can be described by averaging its Hamiltonian over phases of vibrations. Rotationally symmetric conditions on vibrations are assumed in the averaged Hamiltonian. Under these conditions, a bifurcation diagram for the phase portraits of the averaged system is presented. Numerical simulations of different examples of vibrations are performed. The case of proper degeneration in KAM theory guarantees the coherence of dynamical characteristics between the averaged and exact systems.