Bifurcations of a nonlinear spherical pendulum with vibrating suspension point

TL;DR

This study analyzes how high-frequency vibrations of a spherical pendulum's suspension point influence its bifurcation behavior, using averaging methods and numerical simulations to connect the dynamics of the averaged and exact systems.

Contribution

It introduces a bifurcation analysis of a nonlinear spherical pendulum with vibrating suspension, applying averaging and KAM theory to relate averaged and exact system dynamics.

Findings

Bifurcation diagrams for the averaged system are constructed.

Numerical simulations confirm the coherence between averaged and exact dynamics.

Proper degeneration in KAM theory ensures similar behavior in both systems.

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.