The escape rate of asteroids, chemical reaction rates, and fluid mixing rates are all examples of chaotic transport rates in dynamical systems. A Monte Carlo simulation can be used to compute such rates, for example using a model consisting of a system of ODEs or PDEs. The set of trajectories in a chaotic system can be highly complex, and a Monte Carlo simulation often requires millions or billions of trajectories to properly sample the state space and compute accurate transport rates. We study methods of computing transport rates using a smaller number trajectories by studying the structure of the state space or phase space. One way to analyze chaotic phase spaces is to compute symbolic dynamics, which is the labeling of trajectories based on a partitioning of the space. The symbolic dynamics of the system can be represented as a network consisting of a set of partition elements, the nodes, and the allowed transitions between them, the edges. In a Hamiltonian system, for example, a partition element represents a region of phase space, and edges connect pairs of nodes between which transport is allowed in time. A firm grasp of the symbolic dynamics results in the ability to compute important transport rates, including the topological entropy and the escape rate.

One way to compute symbolic dynamics is using invariant manifolds which can divide the state space into pieces. The collection of stable and unstable invariant manifolds is known as a heteroclinic tangle, and the topology of the intersections of stable and unstable manifolds in the tangle encodes information about restrictions on the dynamics. The question we address is How can symbolic dynamics computed from invariant manifolds reduce the number of trajectories required to compute transport rates? In addition, we ask and try to address what useful information does the topology of invariant manifolds tell us about a system that is not apparent from direct or Monte Carlo computation of transport rates? An essential tool in computing the transport rates will be the computation of periodic orbits and using a function called the spectral determinant.

We study several examples of understanding phase space and computing chaotic transport rates using a technique called Homotopic Lobe Dynamics (HLD), which is an automated technique to compute accurate partitions and symbolic dynamics for maps by using the topological forcing by intersections of stable and unstable manifolds of a few anchor periodic orbits. We have applied the HLD technique to analyze and compute transport rates in three systems. In a two-dimensional, double-gyre-like cavity flow that models a microfluidic mixer, we accurately compute the topological entropy over a range of parameter value. In the Hénon map, we use periodic orbits computed from HLD to compute multiexponential decay rates from different zones. In the hydrogen atom in parallel electric and magnetic fields, we use periodic orbits computed from HLD to compute the ionization rate over a range of electron energy where the system exhibits a ternary horseshoe. In each system, computations of transport rates over ranges of parameter value using HLD provided considerable improvements upon previous attempts to compute the same rates.