Subharmonic Resonances and Chaotic Motions of a Bilinear Oscillator

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Abstract
A bilinear oscillator with different stiffnesses for positive and negative deflections arises frequently in off-shore marine technology due to the slackening of mooring lines. A limiting case, in which one of the stiffnesses becomes infinite, is the impact oscillator which has applications to vessels moored in a harbour. The subharmonic resonances, bifurcations and chaotic motions of these oscillators are studied using the concepts of topological dynamics. Problems of the existence, uniqueness and stability of the steady state motions are investigated, and particular use is made of the Poincaré map. The bilinear oscillator is shown to have co-existing small amplitude solutions under most of its subharmonic resonances, showing that one-off and automated computer integrations could easily miss an important resonant peak. The domains of attraction of the competing stable solutions are explored. Cascades of period-doubling bifurcations and the exponential divergence of adjacent starts indicate that the impact oscillator has a régime of chaotic motions governed by a strange attractor.

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