Subharmonic Resonances and Chaotic Motions of a Bilinear Oscillator

A bilinear oscillator with different stiffnesses for positiveand negative deflections arises frequently in off-shore marinetechnology due to the slackening of mooring lines. A limitingcase, in which one of the stiffnesses becomes infinite, is theimpact oscillator which has applications to vessels moored ina harbour. The subharmonic resonances, bifurcations and chaotic motionsof these oscillators are studied using the concepts of topologicaldynamics. Problems of the existence, uniqueness and stabilityof the steady state motions are investigated, and particularuse is made of the Poincaré map. The bilinear oscillatoris shown to have co-existing small amplitude solutions undermost of its subharmonic resonances, showing that one-off andautomated computer integrations could easily miss an importantresonant peak. The domains of attraction of the competing stablesolutions are explored. Cascades of period-doubling bifurcationsand the exponential divergence of adjacent starts indicate thatthe impact oscillator has a régime of chaotic motionsgoverned by a strange attractor.