My research interests center on nonlinear dynamics of dissipative systems. These focus on bifurcation theory, particularly in systems with symmetries, transition to chaos in such systems, low-dimensional behavior of continuous systems and the theory of nonlinear waves. Applications include pattern formation in fluid systems, reaction-diffusion systems, and related systems of importance in geophysics and astrophysics. I am also interested in the theory of turbulent transport and the theory of turbulence.
Pattern formation in large aspect-ratio systems: In many problems in physics it is convenient to ignore the presence of distant boundaries. In the theory of pattern formation in continuous systems (e.g. onset of convection in a fluid heated from below) one finds that even distant boundaries can have surprisingly important effects. This is because the boundaries break continuous symmetries (translations/rotations) present in the unbounded system. I am trying to understand under what conditions can sidewall effects be treated perturbatively (the easy case) and when not (the interesting case). This is particularly relevant when the pattern-forming instability produces propagating waves.
Complex dynamics due to broken symmetries: Even the loss of discrete symmetries can lead to unexpected behavior. For example, the fact that the Earth’s rotation breaks reflection symmetry in planes through the rotation axis is responsible for the pervasive drift of weather patterns. The origin of complex behavior due to forced symmetry breaking can be understood using group-theoretic techniques coupled to bifurcation theory.
Pattern formation in three dimensions: Very little is known about patterns in three dimensions. Currently I am studying the formation of such patterns arising in models of morphogenesis (the Turing instability). Both steady state and time-dependent patterns are being studied using group-theoretic techniques, and related to specific models of the instability.
Strongly nonlinear patterns: In fluid systems with strong restraints (for example rapidly rotating systems, or a strong imposed magnetic field) the flow in one or more directions is inhibited. As a result the motion is simplified and asymptotic techniques can be used to cast the pattern formation problem into the form of a nonlinear eigenvalue problem. In the case of convection the solution of this problem determines the heat flux across the fluid layer for a given applied temperature difference. This approach promises to be extremely useful in studies of geophysical and astrophysical flows which are almost always highly nonlinear, and has already led to a simplified description of rapidly rotating turbulence which agrees well with full 3d simulations and experiments. This simplified but asymptotically exact description has met with a number of successes. For example, it predicts that the 3d turbulent state may be unstable to the formation of large-scale vortices or jets, predictions confirmed by direct numerical simulations of the primitive equations.
Parametric instabilities of vibrating flows: A fluid layer vibrated vertically breaks up into a pattern of surface waves. These waves interact with large scale flows which they themselves generate. A description of this interaction is a challenging task but promises to provide an explanation for the dynamics observed in a number of experiments both on this system and on vibrated soap films.
Dynamics of coupled oscillators: Even very simple systems, such as two coupled nonlinear oscillators, can exhibit dynamical behavior of great complexity, in both relative phase and amplitude. Much of this behavior can be related to global bifurcations in phase space. Systems consisting of many coupled oscillators show even more striking behavior: the oscillators may break up into groups some consisting of coherent or synchronized oscillators and others of incoherent oscillators. These states are called chimera states and the coherent groups may be stationary or propagate through the system.
Spatial localization: Spatially extended, driven dissipative systems frequently exhibit localization. The resulting localized structures can be symmetric or asymmetric and organized in bifurcation diagrams exhibiting a "snakes-and-ladders" structure. This type of structure is robust and this why similar behavior is observed in systems as varied as convection and shear flow, catalytic reactions, optical solitons and even vegetation patterns.