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.

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.