Application of stochastic interrogation to probabilistic models and parametric control

Stochastic interrogation, an experimental method for generating ensembles of transient trajectories, is used to collect data to examine global dynamical behavior. From this method evidence of homoclinic bifurcations can be found. The validity of this method for determining the occurrence of a homoclinic bifurcation is confirmed using a comparison of numerical simulations of the stochastic interrogation method with the numerically calculated stable and unstable manifolds, as well as Melnikov's method.;Stochastic interrogation is also used on a magneto-mechanical oscillator to show a strange non-attracting set. This strange non-attracting set is comprised of an invariant set (a strange saddle which is born following a homoclinic bifurcation), the strange saddle's unstable manifold, and the stable manifold of the attractors for the system. The first experimental images of the evolution of this strange non-attracting set into a strange attractor are presented. The escape rates also show the first experimental evidence of transient times approaching infinity as the system approaches chaos.;Probabilistic models generated from experimental data collected using stochastic interrogation data are constructed. These models, Markov chains, are coarse-grained Frobenius-Perron operators. The phase space is partitioned into equiprobable cells and the transition probability matrix between these cells is constructed. These models accurately capture the global dynamics of the experimental system. Approximations to the natural invariant distribution of a strange saddle are determined from the Markov chain models. The comparison of the rate of loss in transmitted information for Markov chain models from several sets of experimental data is made to show how relative complexity of dynamical behavior can be estimated from these models. The extraction of the escape rate from the probabilistic models is also demonstrated.;A critical numerical investigation of performance limitations of the OGY method for stabilizing unstable periodic orbits within a strange attractor found four factors which can limit performance: observational noise (including digitization noise), choice of surface of section, influence of fractal dimension of the strange attractor, and delay time of the digital feedback loop. A modification of the OGY method is presented to compensate for the delay caused by the feedback loop.