1 Brief Overview

This tutorial explains the application of finite-time Lyapunov exponents (FTLE) for studying time-dependent dynamical systems. The emphasis here is on dynamical systems with arbitrary time dependence, since there is already a nice repertory of tools to tackle time-independent and time-periodic systems.  A leading source for time-dependent dynamical systems are fluid flow problems. So while the ideas stated in this tutorial are expressed in terms of a general dynamical system, we often assume that the system represents a fluid flow.  The evolution of such systems is often governed by partial differential equations, yet it is often acceptable to represent such systems by ordinary differential equations when interest is on large scale transport. This is typically accomplished by either numerically solving an approximation of the Navier-Stokes equation, or taking direct measurements of the fluid. In either case, one typically ends up with with a discrete set of velocity data which represents the vector field of the dynamical system. Therefore, we also emphasize that the given dynamical system might only be known over a finite time-interval.

This tutorial is intended for readers that have had some exposure to dynamical systems concepts, however the presentation is somewhat informal and hopefully easily accessible. Those that have taken a course, or are currently taking a course, in dynamical systems are well-suited to read this tutorial.

In the tutorial, we show that finite-time Lyapunov exponents can be used to find separatrices in time-dependent systems, which are often analogous to stable and unstable manifolds of time-independent systems. These separatrices are called Lagrangian Coherent Structures (LCS).  These structures divide dynamically distinct regions in the flow and reveal geometry which is often hidden when viewing the vector field or even trajectories of the system. Therefore these structures often provide a nice tool in analyzing systems with general time-dependence, especially for understanding transport.

Sections 2 through 6 are mostly dedicated to the theoretical  development of the FTLE and LCS. In Section 7, we demonstrate these concepts on a few examples. Section 8 overviews the algorithmic computation of FTLE fields, and Section 9 provides access to software that can be used to perform these computations. In particular, we specify the steps needed to reproduce results shown in Section 7.3. References are located throughout the tutorial where needed, however Section 10 contains a more coherent discussion of works related to the ideas presented in this tutorial.

Note about movies: Movies are shown in this tutorial as animated gifs which can be started and stopped. To start a movie, single click on the graphic. To stop the movie, single click again on the graphic. On pages which contain multiple movies, there is a shared JavaScript that regulates the start and stop of each movie, therefore if trying to play multiple movies at once, the number of clicks needed to start or stop either of them might get incremented by 1. This is a very minor nuisance, but can be avoided by only viewing one movie at a time.