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. 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. 
Roberto da Silva, Sandra D. Prado, Debora E. Peretti
physics.class-ph