Inverted pendulum driven by a horizontal random force: statistics of the never-falling trajectory and supersymmetry

TL;DR

This paper investigates the statistical properties of a stochastic inverted pendulum driven by horizontal random forces, using supersymmetric field theory, and provides analytical and numerical insights into its never-falling trajectories.

Contribution

It introduces a supersymmetric formalism to describe the statistics of the inverted pendulum's never-falling trajectory under stochastic forcing, offering analytical solutions and numerical validation.

Findings

Statistics of the never-falling trajectory are characterized by the zero mode of a transfer-matrix Hamiltonian.

The mathematical structure resembles a Fokker-Planck equation for the square root of the probability distribution.

Exact analytical solutions are obtained in the strong driving limit, with results matching numerical simulations.

Abstract

We study stochastic dynamics of an inverted pendulum subject to a random force in the horizontal direction (Whitney's problem). Considered on the entire time axis, the problem admits a unique solution that always remains in the upper half plane. We formulate the problem of statistical description of this never-falling trajectory and solve it by a field-theoretical technique assuming a white-noise driving. In our approach based on the supersymmetric formalism of Parisi and Sourlas, statistic properties of the never-falling trajectory are expressed in terms of the zero mode of the corresponding transfer-matrix Hamiltonian. The emerging mathematical structure is similar to that of the Fokker-Planck equation, which however is written for the "square root" of the probability distribution function. Our results for the statistics of the non-falling trajectory are in perfect agreement with direct numerical simulations of the stochastic pendulum equation. In the limit of strong driving (no gravitation), we obtain an exact analytical solution for the instantaneous joint probability distribution function of the pendulum's angle and its velocity.