Our research focuses primarily on the interdisciplinary field of pattern formation, a major branch of nonlinear science. Studies of pattern formation use a common set of fundamental concepts to describe how non-equilibrium processes cause structure to appear in a wide variety of complex systems in nature and in technology. While much progress towards understanding pattern dynamics has been made in recent years, fundamental challenges remain. Characterizing the complex spatiotemporal structures created by physical systems driven out of equilibrium is a huge challenge. Forecasting, which is a central goal in the study of many physical systems, poses another tremendous challenge, and chaos can be a limiting factor to this goal. One well-known example is the weather, illustrated by the so-called butterfly effect: the idea that a small disturbance can be amplified to create drastic, large-scale changes to a system. We approach the challenges of characterization and forecasting by studying turbulence and convection in canonical 2D and 3D fluid flows, such as Kolmogorov flow, Taylor-Couette flow, and Rayleigh-Bénard convection.
In particular, recent computational and theoretical advances have suggested that transitional and weakly turbulent fluid flows can be characterized using special unstable solutions of the Navier-Stokes equations, which govern fluid flows. These unstable solutions are often called “exact coherent structures” (ECS) and have regular temporal behavior (such as equilibria, periodic orbits, etc). We are currently searching for evidence of ECS and exploring their role in guiding turbulent dynamics in Kolmogorov flow and Taylor-Couette flow. This work is a very close collaboration with Roman Grigoriev’s Dynamics and Control Group.
We are also exploring the utility of topological data analysis in characterizing spatiotemporal dynamics. Specifically, we use persistent homology, a powerful tool in algebraic topology, to characterize Rayleigh-Bénard convection. Persistent homology provides a powerful mathematical formalism in which global geometric features of patterns are mapped to points in so-called “persistence diagrams.” This transformation provides a tremendous dimensionality reduction, while preserving the dynamics of the system being probed. This effort is in collaboration with Konstantin Mischaikow’s group (Rutgers) and Mark Paul’s group (Virginia Tech).