Timothy Roberts

My research interests lie in applying geometric ideas to dynamical systems problems. My past work has centered on two main themes: classical ODES, where I used geometric singular perturbation theory to analyze a model a thermoregulation in the brain; and linear spectral theory of traveling fronts and pulses, where I helped develop a new numerical scheme for detecting the stability of fronts and pulses (in one spatial dimension) using geometric ideas (the Riccati transformation). My current work focusses on pattern formation in spatially extended systems, an example of which is given below.

​

Snaking in the Brusselator

​

The Brusselator is one of the oldest systems studied in spatial dynamics, first conceived as a result of Turing's landmark work on the formation of stripe patterns in the 1960's. Despite the decades and myriad studies since then, it remains a system of interest due to its ability to display a zoo of different complex behaviors. In this work we look at a newly discovered behavior, snaking of source defects. Numerical studies by Tzou et al. (2013), found that an instability between two distinct types of stable oscillations allows for the production of source defects: a temporally constant core region sitting in a temporally oscillating background. Their results suggest that these patterns form through a process called snaking. In this work we look to extend their numerical studies with the aim of producing a proof for the existence and stability of these patterns and the snaking bifurcation that produces them.