Classical theory of continuous dynamical systems, defined for all time on a smooth  phase manifold, attempts to classify the dynamical complexity of these systems by identifying asymptotic behavior of their solutions. The asymptotic procedures allow to extract the essential characteristics of such systems based on their behavior in the neighborhood of some lower-dimensional, flow-invariant objects (e.g hyperbolic/elliptic fixed points or periodic orbits, or  more exotic recurrent sets).  However, in the finite-time setting for non-autonomous dynamical systems, when the system is defined or known over a bounded time interval, the classical time-asymptotic notions of hyperbolicity or stability do not apply and new concepts and tools have to be developed in order to study the so called `Lagrangian transport' in such systems.

A major motivation for  this problem arises from the desire to study transport and mixing problems in geophysical flows (e.g., spread of pollutants or phytoplankton) where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field.

In a series of papers with my collaborators

(S. Wiggins, A. D. Kirwan, R. Malek-Madani, A. Mancho) I studied a range of issues associated with the Lagrangian transport in non-autonomous dynamical systems by combining the analytical methods and novel numerical algorithms with both the kinematic models  and/or the output of comprehensive ocean models and measurements. In this approach  the insight into the complex spatio-temporal transport pathways is achieved through the identification of transport barriers (flow invariant manifolds) which govern the global space-time structure of the solutions in the aperiodically time dependent dynamics.