On the stochastic pendulum with Ornstein-Uhlenbeck noise

TL;DR

This paper analyzes the long-term behavior of a frictionless pendulum influenced by Ornstein-Uhlenbeck noise, deriving an analytical energy distribution and demonstrating how noise correlation affects energy fluctuations.

Contribution

It introduces a recursive adiabatic elimination method to derive the asymptotic energy distribution for a pendulum under correlated noise, advancing understanding of stochastic dynamical systems.

Findings

Energy fluctuations are reduced with non-zero noise correlation time.

Analytical energy distribution matches numerical simulations.

Comparison shows the method's advantages over other approximation schemes.

Abstract

We study a frictionless pendulum subject to multiplicative random noise. Because of destructive interference between the angular displacement of the system and the noise term, the energy fluctuations are reduced when the noise has a non-zero correlation time. We derive the long time behavior of the pendulum in the case of Ornstein-Uhlenbeck noise by a recursive adiabatic elimination procedure. An analytical expression for the asymptotic probability distribution function of the energy is obtained and the results agree with numerical simulations. Lastly, we compare our method to other approximation schemes.