The investigator, together with students and colleagues, studies three problems in which temporal or spatio-temporal forcing of oscillatory or excitable systems is important: (1) parametrically excited surface wave patterns, (2) spatio-temporal local feedback in pattern forming systems, and (3) Hopf bifurcation based mechanisms for amplification of sound by inner ear hair cells. Faraday waves, excited on the free surface of a fluid, form in a wide variety of patterns depending on the fluid properties and the form of the periodic forcing function. The investigator's research program focuses on a bifurcation analysis of three- and four-wave interactions when a periodic sequence of delta-function impulses is applied to the fluid container. This idealized forcing function admits unprecedented analytic progress to be made in the linear and weakly nonlinear regimes that apply at or near onset of instability. This project probes how the periodic forcing function may be designed to favor particular patterns. In the second research project spatio-temporal feedback is used to probe the nonlinear pattern formation process, as well as to actively control it. The control of spatio-temporal patterns by local time-delayed and spatially-transformed feedback will be investigated through linear stability analysis, equivariant bifurcation theory, and numerical simulation. In the third project, models of inner ear hair cells, responsible for translating sound-induced motion into electrical signals, are analysed. The initial focus is on amphibian hair cells, for which two separate mechanisms that contribute to the cells' frequency selectivity have been identified - one due to active mechanical motions of the hair bundle and the other captured by an electrochemical model of ion channels in the hair cell body. In each model proximity to a Hopf bifurcation contributes to the amplification properties of the hair cells. The investigator's research project uses dynamical systems methods to derive a reliable reduced model, from existing detailed physiological models of the two Hopf bifurcation mechanisms, with attention to the effects of this two-stage amplification on gain and frequency selectivity. This project lays the foundation for further investigation of the effects of coupling the hair cell bundles.
Many spatially extended nonlinear systems, including hydrodynamic and laser systems, exhibit spatio-temporal chaotic behavior when subjected to external forcing. The investigator's research program will lead to a deeper understanding of how to eliminate irregular behavior in favor of spatio-temporally regular patterns. This is done through appropriate design of the temporal forcing function in the case of hydrodynamic waves, or through spatio-temporal feedback in the case of nonlinear optical and chemical systems. Careful comparison between theoretical results and results of experimental investigations will be made, providing valuable feedback to this research effort. The investigator's analysis of biophysical models of inner ear hair cells contributes to a greater understanding of how the nonlinearities in two proposed mechanisms of frequency selectivity and amplification might work together to achieve greater gain. The training of applied mathematics graduate students and postdoctoral fellows in interdisciplinary research activities is an integral part of the research effort.
