Graduate course on dynamical systems

Prize winner, Teaching DS Competition, 2013

Course materials for two-quarter (20 week) graduate course on dynamical systems. (Advanced undergraduates have successfully taken the course.) The courses integrate nonlinear dynamics (low- and high-dimensional systems), symbolic dynamics, information theory, and computation theory. The theme, though, is to show how dynamical systems has (radically) modified how we do science.

Emerging Behavior and Spatiotemporal Chaos in Reaction-Diffusion Models: GPU-accelerated simulations in a web browser over the internet

A set of interactive programs to study and analyze several models of excitable media in tissue. As the waves they produce propagate through the media, the models exhibit complex spatiotemporal dynamics that cannot be appreciated from an analysis of the underlying equations or even verbal descriptions.

Here, we allow users to perform in real time simulations of these models and to watch the patterns develop and change over time as the simulated dynamical waves propagate. The parameters governing the model's behavior can be changed on the fly to alter the dynamics. In addition, users can apply perturbations and periodic pacing, that change the patterns locally an globaly and watch the response.

The educational software we worked on is based on WebGL. That is, programs that run over the web on a browser, yet, they are very-very fast as they run on the GPU instead of the CPU.

Head to the website below to get started!

The Self-Driven Particle Model

The Self-Driven Particle Model is a toy dynamical system in which particles move in 2-dimensions, and interact with each other according to a simple rule. Particles move at a constant speed, and their orientation is set to be the average orientation of all particles (including themselves) within an interaction radius, plus a random term. This model has been shown to exhibit complex dynamical behavior, including a 2nd order phase transition, criticality and clustering. This tutorial introduces the model incrementally, and depicts the computation of the order parameter, critical parameter and critical exponent.

Space Travel: Mathematics Uncovers an Interplanetary Superhighway

Contrary to everyday experience on Earth, the most efficient route through space may not be a straight line. Some mathematicians and NASA engineers have learned in recent years that take best advantage of gravity, and save fuel in the process, it may be necessary to make bizarre loops through space.

Renormalization and Scaling in Applied Mathematics

This tutorial is based upon lectures that were given in Bonn-Rottgen, Germany during August 2004. Approximately thirty participants attended this Summer School that was made possible by a grant of the German Research Foundation (DFG) entitled: Priority Program 1095 "Analysis, Modeling and Simulation of Multiscale Problems," and organized by Dr. Christof Eck and Prof. Heike Emmerich.