RESEARCH DESCRIPTION
My area of research is Dynamical Systems. I am interested in all types of dynamical phenomena. My goals are to explain, describe, and ultimately predict time evolutions of dynamical processes. A large part of my work has to do with understanding chaotic behavior. I have worked with (i) general theory (called nonuniform hyperbolic theory) and (ii) analyses of concrete systems. My favorite topics in (i) include Lyapunov exponents, entropy and dimension, SRB measures for strange attractors, random perturbations and zero-noise limits, rates of correlation decay and large deviations for dynamical observations, as well as leaky systems. Concrete models that I have worked with include billiards (uniform motion of particles with collisions) and periodically kicked oscillators. A topic straddling (i) and (ii) is low dimensional strange attractors (in phase spaces of arbitrary dimensions). They appear naturally following a system's loss of stability, as in, e.g., shear-induced chaos. In recent years, I have given increasing emphasis to systems that are large, driven, or out of equilibrium, recognizing that such models are closer to real-world systems. Examples of large systems include semiflows defined by PDEs and complex networks. Driving forces may be deterministic or stochastic, or the system may be in contact with (unequal) heat baths. My goals here are to identify dynamical mechanisms behind observed phenomena and to discover emergent behaviors, in nonequilibrium steady states and in a system's responses to stimuli. Additionally I have become interested in applications of dynamical systems ideas to the sciences and engineering. Two disciplines with which I have tried to make contact are nonequilibrium statistical mechanics and theoretical neuroscience. I am currently working on the latter, studying the dynamics of mammalian visual cortex via computational modeling. |