Stability of Multi-Degree of Freedom Stochastic Nonlinear Systems

Melnikov's method for nding necessary conditions for homoclinic chaos is extended to a class of stochastically forced multi-degree of freedom nonlinear systems. The stochastic forcing induces a stochastic Melnikov process, and simple zeros of this process imply transversal intersection of appropriate stable and unstable manifolds. This process can also provide information on probabilities of escape and mean exit rates from regions of phase space. In generalizing results from single degree of freedom stochastic systems, however, some new concepts (and reinterpretations of old concepts) are addressed. Applications are presented to the dynamics of a feedback controlled buckled column subject to stochastic forcing. The stochastic Melnikov process is calculated, and illustrations of the e ect of system parameters on system dynamics and chaos are given. The use of the Melnikov process as an estimator of mean exit rates from preferred regions of phase space is investigated.