Spiral and target patterns are ubiquitous features in systems driven by diffusion. They occur in physiological systems, such as the electrochemical signals of the heart; physical systems, such as oxidation layers on platinum alloys; and, biological systems such as populations of slime mold. Despite this prevalence, and the study of pattern forming systems over the last two decades, understanding the existence and dynamics of these patterns pose a significant challenge. To illustrate this I will present new results on the existence and bifurcation of 1-dimensional spiral and target patterns in reaction-diffusion equations. In this work we prove that spirals and targets can be formed simultaneously in very complex and intricate bifurcations diagrams in a process called snaking, and how this process can be understood using a dynamical systems perspective.

Bio: Timothy Roberts is a William H. Kruskal Instructor in the Department of Statistics at the University of Chicago. He completed his PhD in Applied Mathematics at Brown University with his Advisor Bjorn Sandstede in 2024. His research focuses on the formation of patterns in PDE models inspired by biological systems, and using rigorous mathematical techniques (such as spatial dynamical systems and singular perturbation theory) to study their existence, stability and bifurcation.

See all upcoming talks at https://www.anl.gov/mcs/lans-seminars