My research is broadly in the area of dynamical systems. I study applications of bifurcation theory to 'tipping points' and to spontaneous pattern formation in physical systems. Currently I am especially interested in dynamical systems models related to Earth's climate and ecosystems. My focus is on understanding the mathematical mechanisms behind qualitative changes in system behavior with changes in system parameters. Also of interest are mathematical tipping point mechanisms that involve noise and periodic forcing, or that result from drifting parameters, which may lead to 'rate-induced tipping' in a non-autonomous dynamical systems setting. I want to understand the robustness of these mechanisms to changes in the mathematical model of a problem; this could inform efforts to develop 'early warning signs' of abrupt transitions that may occur as system parameters drift.
In the past I have investigated spatio-temporal pattern formation in driven fluid dynamical systems, including mechanisms for parametrically exciting the formation of superlattice and quasi-patterns, as well as spatio-temporal chaos, as observed in laboratory experiments. I have also investigated methods of feedback control of unstable periodic orbits that relied on incorporating time delays, extending that approach to stabilization of some unstable nonlinear traveling waves and to patterns of oscillation for symmetrically coupled nonlinear oscillators. Some of the mathematical themes of my work include equivariant bifurcation theory applied to spontaneous symmetry-breaking, global bifurcations and heteroclinic networks in equivariant dynamical systems, bifurcation theory for delay differential equations and for piecewise-smooth dynamical systems, and dimension reduction approaches for investigating dynamical systems that possess a natural time scale separation.