Lagrangian Coherent Structures: Analysis of time-dependent dynamical systems using finite-time Lyapunov exponents

Third Prize, DSWeb Tutorials Contest

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.

Excitable Media (Java Applets)

Examples of Excitable media in 0D, 1D and 2D with emphasis to cardiac dynamics. The page contains more than 40 Java Applets dedicated to teach the origen of excitability, as well as the dynamics and stability of waves in 1D and spiral waves in 2D.

Crisis-induced Intermittency in Coupled Chaotic Maps

Honorable Mention, DSWeb Tutorials Contest

Intermittent transitions between multiple dynamical states are characteristic nonlinear phenomena in dynamical systems. It is important to understand a mechanism of an onset of intermittency in mathematical models, because it often replicates an observable phenomenon in physical world. Among several types of intermittency, we focus on crisis-induced intermittency in this tutorial.

In coupled chaotic maps which are widely used in modeling networks of dynamical elements, coexistence of multiple attractors is not uncommon. As a system parameter is varied, such a system often exhibits a sudden crisis inducing intermittent behaviors. We demonstrate that a crisis can be understood by a contact between attractors and a fractal basin boundary as well as an emergence of a snap-back repeller in two coupled chaotic maps. The scenario for the crisis is also illustrated with qualitative changes in basin structure and quantitative changes of fractal dimension of the basin boundary.

The approaches to the crisis is successfully applied to an analysis of a global bifurcation inducing itinerant memory dynamics in a chaotic neural network. In addition to the viewpoints for crisis-induced intermittency, this tutorial provides several basic concepts such as invertible and non-invertible maps, smooth and fractal basin boundaries, fractal dimension, and basin bifurcations in discrete-time dynamical systems.

Community of Ordinary Differential Equations Educators

The Community of Ordinary Differential Equations Educators (CODEE) seeks to improve the teaching and learning of ordinary differential equations. One way to do this is to increase student engagement and active learning is through the use of projects involving modeling and computer experiments. So CODEE, with the support of the National Science Foundation (1992 – 97 and 2008 – 2013), has constructed a digital library which supports this purpose (www.codee.org). The digital library contains the CODEE Journal which contains articles about the teaching and learning of ODEs, as well as classroom-ready projects, and reviews of other teaching materials. The CODEE web site also describes some software that supports the teaching and learning of ODEs, as well as archives of the newsletters and other materials produced during the first NSF grant. The ultimate goal of the digital library is to create a community for instructors to find, share, and discuss resources for teaching ordinary differential equations. To learn more about submitting a manuscript to the CODEE digital library, visit the Editorial Policy page of the CODEE web site.

A tutorial on KAM theory

This is a tutorial on some of the main ideas in KAM theory. The goal is to present the background and to explain and compare somewhat informally some of the main methods of proof. It is an expanded version of the lectures given by the author in the AMS Summer Research Institute on Smooth Ergodic Theory Seattle, 1999. The style is pedagogical and expository and it only aims to be an introduction to the primary literature. It does not aim to be a systematic survey nor to present full proofs. It only covers the classical results and not the most modern ones. Comments and suggestions for improvement are welcome. We are planning a revised and expanded version.