My vita (last updated March, 1999) (PDF format)
Research interests: Plasma theory and nonlinear dynamics
I am most interested in dynamics, i.e. the study of how things change. I am particularly interested in the development of new tools which will help us to peer more deeply and with more subtlety into the dynamical behavior of physical systems. I have done work on soliton theory, turbulence, signal processing, and short wavelength asymptotics. Much, but not all, of this work has dealt with nonlinear phenomena. My present research interests fall into two broad categories:
The current emphasis of my work in nonlinear dynamics concerns the analysis of signals from nonlinear systems. One major area of effort concerns the use of symbolic methods to analyze signals. In this approach the signal is severely coarse grained and converted into a long string of just a few symbols (for example, '0' and '1'). The statistical properties of this symbol string are then analyzed for multi-step correlations. Such an approach to the analysis of signals is appealing because: 1] it is robust to noise, and 2] it involves only counting in binary and is, hence, very fast computationally. Previous work by our group and others have shown that the symblic data alone is often sufficient for solving many of the standard problems in signal analysis: estimation of characteristic timescales, detection of periodicities, and the testing and validation of models. Our current interests concern the detection of symbolic precursors of instabilities. Here the dynamics of the system is weakly non-stationary in time. The symbolic data is sampled in moving windows and the statistical properties of the symbols in different windows is compared. In the figure at right we show a cross correlation plot comparing the symbolic behavior of a dynamical system with the noise driving it. The system eventually undergoes what is called a subcritical Hopf bifurcation (thought to occur, for example, in turbines and other machines). Early times are at the upper left and later times at the lower right. The instability occurs along the right edge. What should be noticed is the shifting pattern of color (implying changing statistical behavior) as one moves to the right (toward the instability).
Phase space analysis of wave propagation in non-uniform media
This topic (also known as ray tracing or WKB analysis) has a long history in mathematical physics, dating all the way back to the original work of Hamilton who first derived his famous equations in the context of wave optics in nonuniform media. Short wavelength asymptotics has become a well-developed area of mathematical physics and applications can be found in all branches of physics. My efforts in this area have concentrated on situations where the WKB approximation breaks down. This work is carried out in collaboration with Prof. Allan N. Kaufman of UC Berkeley and the plasma theory group at Lawrence Berkeley National Lab.
For example: Non-uniform plasmas can support many different types of waves and in some situations there can be regions where two or more waves become "degenerate" (both local dispersion relations are satisfied simultaneously). This is a resonance phenomena, much like the coupling of two oscillators, and it leads to a significant exchange of energy, momentum, and action between the various wave modes. In these degenerate regions the WKB method fails and a local approximation must be used. We have recently developed a general set of analytical tools which cast these problems into a canonical form for which a general solution can be found. We are extending these techniques to include sources in order to study the emission of radiation by particles in nonuniform media, a basic problem which is still open. The firgure at right shows a phase space diagram for a ray analysis of RF heating in a tokamak. This involves launching electromagnetic (magnetosonic) rays from an antenna, propagating them into the plasma and through resonance with another mode (the ion-hybrid). As this resonance is crossed, some of the energy of the incoming waves is converted to the ion-hybrid, and some transimitted through the resonance. Because the magnetosonic rays are confined they bounce at the edge of the plasma and can re-enter the resonance. This can occur ad infinitum with the energy gradually leaking into the ion-hybrid mode and propagating to high wavenumbers, where it generates fine scale spatial structure in the plasma which is damped by other processes. We have also recently applied these techniques to the study of equatorial wave dynamics in the ocean.
Recent talks and publications
Other activities: The Graduate Center at William & Mary