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Deterministic and stochastic aspects of the stability in an inverted pendulum under a generalized parametric excitation. (English) Zbl 1443.70022

Summary: 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.

MSC:

70E17 Motion of a rigid body with a fixed point
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