Lyapunov exponent for Whitney's problem with random drive

TL;DR

This paper analyzes the statistical behavior of a non-falling inverted pendulum driven by random forces, extending previous work to finite times and calculating the Lyapunov exponent to understand correlation decay.

Contribution

It generalizes the transfer-matrix approach to finite time intervals and multipoint correlations for the Whitney problem with stochastic driving.

Findings

Derived finite-time correlation functions.

Calculated the Lyapunov exponent for the system.

Extended supersymmetric field theory methods.

Abstract

We consider the statistical properties of a non-falling trajectory in the Whitney problem of an inverted pendulum excited by an external force. In the case when the external force is white noise, we recently found the instantaneous distribution function of the pendulum angle and velocity over an infinite time interval using a transfer-matrix analysis of the supersymmetric field theory. Here, we generalize our approach to the case of finite time intervals and multipoint correlation functions. Using the developed formalism, we calculate the Lyapunov exponent, which determines the decay rate of correlations on a non-falling trajectory.