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.

In many problems in dynamical systems one is interested in the identification of sets which have qualitatively different fates. The finite-time Lyapunov exponent (FTLE) method is a general and equation-free method that identifies codimension-one sets which have a locally high rate of stretching around which maximal exponential expansion of line elements occurs. These codimension-one sets thus act as transport barriers. This geometric framework of transport barriers is used to study various problems in phase space transport, specifically problems of separation in flows that can vary in scale from the micro to the geophysical. The first problem which we study is of the nontrivial motion of inertial particles in a two-dimensional fluid flow. We use the method of FTLE to identify transport barriers that produce segregation of inertial particles by size. The second problem we study is the long range advective transport of plant pathogen spores in the atmosphere. We compute the FTLE field for isobaric atmospheric flow and identify atmospheric transport barriers (ATBs). We find that rapid temporal changes in the spore concentrations at a sampling point occur due to the passage of these ATBs across the sampling point. We also investigate the theory behind the computation of the FTLE and devise a new method to compute the FTLE which does not rely on the tangent linearization. We do this using the 925 matrix of a probability density function. This method of computing the geometric quantities of stretching and FTLE also heuristically bridge the gap between the geometric and probabilistic methods of studying phase space transport. We show this with two examples. / Ph. D.