Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation

Abstract

The present paper outlines a general second-order dynamical system on manifolds and Lie groups that leads to defining a number of abstract non-linear oscillators. In particular, a number of classical non-linear oscillators, such as the simple pendulum model, the van der Pol circuital model and the Duffing oscillator class are recalled from the dedicated literature and are extended to evolve on manifold-type state spaces. Also, this document outlines numerical techniques to implement these systems on a computing platform, derived from classical numerical schemes such as the Euler method and the Runke-Kutta class of methods, and illustrates their numerical behavior by a great deal of numerical examples and simulations.

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“…In case the manifold is a Lie group, the expression of a general second-order system on a Lie group becomes [39]…”