Chaos

My work in chaos started at Cornell with my doctoral study on an oscillator with dry friction.  At the root of this oscillator's dynamics was a simple iterated map much like the famous logistic (quadratic) map, a paradigm of chaos analyzed by the chaos pioneers who discovered period doubling and other universal behavior.  

Chaos can be studied from an established model, either by analytical treatment or simulation of the model.  Steve Shaw and I teach a course (ME 961) with this perspective, involving local bifurcation theory, global bifurcations due to homoclinic tangles, symbol dynamics and the horseshoe map, and perhaps routes to chaos (period doubling, intermittency, torus wrinkling, torus doubling, and quasiperiodicity), or "codimension-2" bifurcations.  A related course is ME 863 (nonlinear oscillations), which focuses less on chaos and more on nonlinear resonances and self-excitation.  Chaos can also be studied by analyzing data from a chaotic process.  When I get a chance, I do a course (ME 960) on the treatment of chaotic data.  The idea is to take measurements of a chaotic process and recreate the geometry of what's actually going on.  Ultimately, the process can be characterized by its topology, fractal dimension, Lyapunov exponents, and entropy.  Methods of "chaos theory" can be used to distinguish deterministic from random processes, uncover a model of the process, and lead to nonlinear prediction. 

Chaos applications include, for example, the mixing on non-turbulent fluids, coding and communication, system identification, and control  Chaos is advantageous for system identification because chaotic data has more information than periodic data.  Such dynamical information can be exploited to obtain concrete information, such as parameter values, about the system at hand.  Finally, the reality is that nonlinear phenomena are present in a vast number of interesting systems.  Understanding chaos is essential for understanding the behavior of many of these nonlinear systems.  Chaos theory allows us to know the way things work.

The papers below deal with descriptions of chaotic behavior and nonlinear phenomena, developments of chaos tools, and modeling.

Support:  NASA Langley Research Center, 4/01-11/02 (Parametric System Identification), National Science Foundation CMS-9624347 (CAREER, primarily on Friction Dynamics), 8/96-7/00. (Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.)

Publications involving chaos