Deterministic and stochastic aspects of the stability in an inverted pendulum under a generalized parametric excitation

TL;DR

This paper investigates the stability of an inverted pendulum under complex parametric excitations, combining numerical simulations and analytical methods to understand deterministic and stochastic stabilization effects across different excitation complexities.

Contribution

It introduces a simple stability condition applicable to multiple excitation cases and compares numerical and analytical results, revealing broader stability regions and optimal excitation parameters.

Findings

Numerical simulations show wider stability regions than analytical predictions.

Stochastic noise can enhance the stability of the inverted pendulum.

Optimal number of excitation cosines maximizes survival probability.

Abstract

In this paper, we explore the stability of an inverted pendulum under a generalized parametric excitation described by a superposition of $N$ cosines with different amplitudes and frequencies, based on a simple stability condition that does not require any use of Lyapunov exponent, for example. Our analysis is separated in 3 different cases: $N=1$, $N=2$, and $N$ very large. Our results were obtained via numerical simulations by fourth-order Runge Kutta integration of the non-linear equations. We also calculate the effective potential also for $N>2$. We show then that numerical integrations recover a wider region of stability that are not captured by the (approximated) analytical method. We also analyze stochastic stabilization here: firstly, we look the effects of external noise in the stability diagram by enlarging the variance, and secondly, when $N$ is large, we rescale the amplitude by showing that the diagrams for time survival of the inverted pendulum resembles the exact case for $N=1$. Finally, we find numerically the optimal number of cosines corresponding to the maximal survival probability of the pendulum.