Bifurcations of phase portraits of pendulum with vibrating suspension point

TL;DR

This paper analyzes how rapid vibrations of a pendulum's suspension point affect its phase portrait bifurcations, providing a comprehensive description of the resulting dynamical changes in the averaged Hamiltonian system.

Contribution

It offers a complete classification of bifurcations in the phase portraits of a pendulum with vibrating suspension point, based on an averaged Hamiltonian model.

Findings

Bifurcation diagrams of the averaged system are fully characterized.

Conditions for different bifurcation types are explicitly derived.

The analysis enhances understanding of vibrational stabilization and control of pendulum dynamics.

Abstract

We consider a simple pendulum whose suspension point undergoes fast vibrations in the plane of motion of the pendulum. The averaged over the fast vibrations system is a Hamiltonian system with one degree of freedom depending on two parameters. We give complete description of bifurcations of phase portraits of this averaged system.