UC Berkeley

In this thesis, a novel investigation of deterministic nonlinear systems is presented. When autonomous dynamical systems are randomized by white-noise excitation, their behaviors are governed by diffusion equations. The core idea behind the method of randomizationinvolves replacing a nonlinear ordinary differential equation of motion by a linear partial differential equation of diffusion.

Exact analytical solutions are feasible for only a limited number of nonlinear systems. However, some nonlinear systems, which are difficult to analyze using deterministic methods, possess stationary diffusion equations with exact solutions. These diffusion responses provide deeper insights into the qualitative behaviors of nonlinear systems, such as stability at multiple equilibria, limit cycles, and bifurcations.

To demonstrate the potential and feasibility of randomization, numerous nonlinear oscillators are considered. When both the deterministic systems and the associated randomized systems can be analyzed, there is complete agreement in their qualitative properties. Furthermore, some example systems inaccessible through deterministic approaches can be readily examined using randomization, suggesting that randomization could be an alternative tool for investigating nonlinear oscillations.