| This dissertation presents computationally efficient methods for numerical computation of conditional probability in a nonlinear dynamical system. That is, if the probability density of a system within its state space is assumed to be known at one instant, the probability that the state will satisfy a certain criterion within a set period is found. Nonlinear dynamical systems are deterministic, however the nonlinearities of the systems make them unpredictable with finite computational techniques. This unpredictable behavior promotes probabilistic studies. Two classes where stochastic behavior in a bounded system arise are examined, the Hamiltonian and the dissipative.; For Hamiltonian systems, an algorithm derived from the Frobenious-Perron operator and the Liouville equation is presented. For chaotic dissipative systems, a method in which probabilistic pre-images are pre-computed is developed, implemented and tested. The probabilistic pre-images are calculated by determining the regions of state space that have points which will evolve to enter an area of interest within a certain period. For each of these regions, the conditional probability that a random point will enter the area of interest can be found and tabulated. This tabulated data then may later be used by real-time systems to calculate an overall conditional probability by summing the conditional probabilities of those particular regions which comprise the initial probability distribution.; To show the utility of the method, numerous numeric comparisons of this probabilistic pre-image technique to Monte-Carlo simulations are performed. Resulting conditional probability is compared for both 2-dimensional and 3-dimensional systems. Additionally, analytic and experimental calculations of the numeric complexity are obtained.; The experimental data confirm that when compared to a Monte-Carlo simulation, the probabilistic pre-image method markedly reduces the computations needed to compute conditional probabilities in nonlinear dynamical systems. The computational savings are directly proportional to the duration of the dynamical system for which the conditional probability is computed. In most cases, the conditional probabilities computed via probabilistic pre-image retain accuracies within five percent of the Monte-Carlo results. |
