Nonlinear Phenomena in Complex Systems, 1999, Vol.2, No.3, pp.1--13

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

1999, Volume 2, Number 3, pp.1--13

Subharmonic Bifurcations and Melnikov's Theory in a System of Parametrically Driven Pendulums.
T. Bountis and J.P. Van Der Welle

Melnikov's theory of homoclinic bifurcations has proved to be a most successful tool for predicting the onset of (homoclinic) chaos in periodically perturbed dynamical systems of two or more dimensions. By contrast, Melnikov's predictions for the occurrence of subharmonic bifurcations have not been as widely tested and applied to many-dimensional systems of physical significance. In this paper we first demonstrate explicitly the remarkable accuracy of Melnikov's subharmonic theory in predicting the appearance of in-phase and counter-phase oscillations, as well as in-phase rotations, in a system of two damped, parametrically driven pendulums. Secondly, we propose a generalization of Melnikov's approach by which more complicated motions can be studied in coupled pendulum systems of 2N phase space dimensions.

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