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

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 of the effective potential. 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 survival time 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.