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. / Ph. D. / The prediction and control of critical events in engineering systems has been a major objective of scientific research in recent years. The multifaceted problems facing the modern society includes critical events such as spread of pathogens and pollutants in atmosphere and ocean, capsize of boats and cruise ships, space exploration and asteroid collision, to name but a few. Although, at first glance they seem to be disconnected problems in different areas of engineering and science, however, they have certain features that are inherently common. This can be studied using the abstraction of phase space which can be thought of as the universe where all possible solutions of the governing equations, derived using principles of physics, live and evolve in time. The <i>phase space</i> can be just 2D, 3D or even infinite dimensional but the critical events manifest themselves as volumes of phase space, which represent solutions at a given instant of time, get transported from one region to another due to the underlying dynamics. This mathematical abstraction is called phase space transport and studied under the umbrella of dynamical systems theory. The geometric view of the solutions that live in the phase space provides insight into the mechanisms of how the critical events occur, and the understanding of these mechanisms is useful in deciding about control strategies.
A slightly different view for understanding critical events is to consider a thought experiment where a ball is rolling on a multi-well surface or potential well. As the time evolves, the ball will escape from its initial well and roll into another well, and eventually start exploring all the wells in a seemingly unpredictable way. However, these unpredictable escape/transition can be studied systematically using methods of chaos and dynamical systems. The escape/transition in a potential well implies a dramatic change in the behavior of the system, and hence the significance in prediction and control of <i>escaping dynamics</i>. The control aspect becomes more challenging due to inherent disturbance in the system that is difficult to model and we may not have the equal or more control authority to cancel those disturbances. However, we can usually estimate the maximum values of the disturbance, and try to avoid escaping from the potential well while using a smaller control. This idea is called <i>partial control of escaping dynamics</i> and can guarantee avoidance of escape for <i>ad infinitum</i>.
In this doctoral research, we focus on the two mechanisms, phase space transport and escaping dynamics, by considering problems from fluid dynamics and capsize of a ship. The applications are used for numerical demonstration and evidence of the general approach in studying a large class of problems in classical physics.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/73364 |
| Date | 01 November 2016 |
| Creators | Naik, Shibabrat |
| Contributors | Engineering Science and Mechanics, Ross, Shane D., Paul, Mark R., Jung, Sunghwan, Woolsey, Craig A., Puri, Ishwar K. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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