Prof. Elizabeth Bradley, University of Colorado is visiting the Department of Information and Computer Science, hosted by Jaakko Hollmén. Bradley is an editor of the journal Chaos: An Interdisciplinary Journal of Nonlinear Science and has been program chair of Dynamics Days in 2006, International Workshop on Qualitative Reasoning in 2008, and the International Symposium on Intelligent Data Analysis in 2003 and in 2011.
On 24th of October, Bradley gave an invited talk entitled "Chaos and Control". She provided a review of the mathematical theory and computational techniques that are used in the control of chaos, and covered a variety of examples ranging from science and engineering to music and dance.
Bradley started by introducing basic concepts related to the dissipative dynamical systems. Chaos can be defined as complex behavior, arising in a deterministic nonlinear dynamic system, which exhibits two special properties: (1) sensitive dependence on initial conditions, and (2) characteristic structure. Properties of chaotic or "strange" attractors include neighboring trajectories diverge exponentially, covered densely by trajectories.
Bradley emphasized that chaos is not an academical oddity. Actually, nonlinearity and chaos are ubiquitous, e.g., in hearts, brains, populations, planets, black holes, pulsars, flows of heat and fluids, and many other kinds of systems. Key concepts of chaotic systems from the point of view of control include
- characteristic attractor geometry,
- exponential trajectory separation,
- dense attractor coverage,
- un/stable manifold structure, and
- unstable periodic orbits
From the point of view of problem solving, denseness indicates reachability. In other words, trajectories on a chaotic attractor densely cover a set of non-zero measure and thus make all points in that set reachable from any initial condition in its basin of attraction. Reachability is nondeterministic and therefore using chaos in control is not for time-critical applications. Bradley told about an early instance of research related to this topic, published as "Using Chaos to Broaden the Capture Range of a Phase-Locked Loop" (1993). Bradley discussed how to target a specific point on the attractor, exploiting sensitive dependence on initial conditions for control leverage and controllability. Here she referred to her MIT PhD thesis "Taming Chaotic Circuits" (1993) and to Troy Shinbrot and his colleagues' work. Bradley also warned about being naive in one's expectations regarding applications of controlling complex chaotic systems. For instance, it is impossible to find out where to make a butterfly to flap its wings in order to control the future development and route of a hurricane.
Towards the end of her presentation, Bradley discussed two applications of chaos theory in art. First, she introduced Diana Dabby's work on generating variations using chaotic mapping. A convincing example was a variation of J.S. Bach's Prelude in C from the Well-tempered Clavier, Book I generated by the chaotic mapping. The details of the mapping technique are described in the article "Musical variations from a chaotic mapping". As a second application in the area of performing arts, Bradley described her and her collaborators' work on dance. She discussed, for example, the paper "Learning the Grammar of Dance" by Joshua Stuart and herself.