Geometric Approaches in Phase Space Transport and Partial Control of Escaping Dynamics

Abstract

This dissertation presents geometric approaches of understanding chaotic transport in phase space that is fundamental across many disciplines in physical sciences and engineering. This approach is based on analyzing phase space transport using boundaries and regions inside these boundaries in presence of perturbation.

We present a geometric view of defining such boundaries and study the transport that occurs by crossing such phase space structures. The structure in two dimensional non-autonomous system is the codimension 1 stable and unstable manifolds associated with the hyperbolic fixed points. The manifolds separate regions with varied dynamical fates and their time evolution encodes how the initial conditions in a given region of phase space get transported to other regions. In the context of four dimensional autonomous systems, the corresponding structure is the stable and unstable manifolds of unstable periodic orbits which reside in the bottlenecks of energy surface. The total energy and the cylindrical (or tube) manifolds form the necessary and sufficient condition for global transport between regions of phase space.

Furthermore, we adopt the geometric view to define escaping zones for avoiding transition/escape from a potential well using partial control. In this approach, the objective is two fold: finding the minimum control that is required for avoiding escape and obtaining discrete representation called disturbance of continuous noise that is present in physical sciences and engineering. In the former scenario, along with avoiding escape, the control is constrained to be smaller than the disturbance so that it can not exactly cancel out the disturbances.