Stability analysis of the nonlinear pendulums under stochastic perturbations

TL;DR

This paper analyzes the stability and bifurcations of a nonlinear pendulum subjected to stochastic vibrations, using averaging techniques and numerical methods to understand the system's behavior under randomness.

Contribution

It provides a complete bifurcation analysis of the averaged Hamiltonian system and links it to the exact stochastic system through numerical and theoretical methods.

Findings

Bifurcation curves of the stochastic system are numerically characterized.

Averaging simplifies the analysis of stochastic perturbations.

The correspondence between averaged and exact systems is established via Poincaré maps.

Abstract

We consider a nonlinear pendulum whose suspension point undergoes stochastic vibrations in its plane of motion. Stochastic vibrations are constructed by stochastic differential equations with random periodic solutions. Averaging over these stochastic vibrations can be simplified with ergodicity. We give a complete description of the bifurcations of phase portraits of the averaged Hamiltonian system. The bifurcation curves of the stochastic perturbed Hamiltonian system are shown numerically. Estimations between the averaged system and the exact system are calculated. The correspondence of the averaged system to the exact system is explained through the Poincar\'e return map. Studying the averaged Hamiltonian system provided important information for the exact stochastic perturbed Hamiltonian system.