Contributions to the computational analysis of multi-dimensional stochastic dynamical systems

Several contributions in the area of computational stochastic dynamics are discussed; specifically, the response of stochastic dynamical systems by high order closure, the response of Poisson and Gaussian white noise driven systems by solution of a transformed generalized Kolmogorov equation, and control of nonlinear systems by response moment specification.; Statistical moments of response are widely used in the analysis of stochastic dynamical systems of engineering interest. It is known that, if the inputs to the system are Gaussian or filtered Gaussian white noise, Itô's rule can be used to generate a system of first order linear differential equations governing the evolution of the moments. For nonlinear systems, the moment equations form an infinite hierarchy, necessitating the application of a closure procedure to truncate the system at some finite dimension at the expense of making the moment equations nonlinear. Various methods to close these moment equations have been developed. The efficacy of cumulant-neglect closure methods for complex dynamical systems is examined.; Various methods have been developed to determine the response of dynamical systems subjected to additive and/or multiplicative Gaussian white noise excitations. While Gaussian white noise and filtered Gaussian white noise provide efficient and useful models of various environmental loadings, a broader class of random processes, filtered Poisson processes, are often more realistic in modeling disturbances that originate from impact-type loadings. The response of dynamical systems to combinations of Poisson and Gaussian white noise forms a Markov process whose transition density satisfies a pair of initial-boundary value problem termed the generalized Kolmogorov equations. A numerical solution algorithm for these IBVP's is developed and applied to several representative systems.; Classical covariance control theory is extended to the case of nonlinear systems using the method of statistical linearization. The design procedure is applied to several nonlinear systems of civil engineering interest including hysteretic oscillators. The idea of covariance control is then generalized to the problem of response moment specification where higher order response moments are prescribed with the hope of having more authority over response extremes. The algorithm is then demonstrated by application to a Duffing oscillator.