Math and Climate
Conley Index Theory for Nonsmooth and Set-Valued Dynamical Systems
This page covers the research that I did for my PhD under Richard McGehee
Schematic of Welander's ocean box model
Climate Models with Sudden Jumps
Conceptual climate models often feature some sort of sudden change in the system. In oceanography, for example, models often split the ocean into deep region and shallow regions. In the schematic on the left, the dotted line represents the boundary between the shallow and deep ocean regions.
A typical piecewise-continuous vector field in the plane with a discontinuity boundary on the x-axis.
Differential Equations with Discontinuities
When physical systems feature instantaneous changes, like the ocean box model to the left, the vector field describing its dynamics is often discontinuous. That means we can't rely on a lot of the theorems that we use to analyze traditional differential equations. My research aims to improve the foundations of these systems by generalizing Conley Index Theory to this setting.
Conley Index Theory is a tool that describes the behavior of a dynamical system that is completely robust under perturbation. The theory gives us a qualitative understanding of isolated invariant sets and the structure of their attractors. It sidesteps the problems associated with bifurcation theory by focusing on rough, topological features of the system. For many models, particularly models which aim to describe extremely complicated systems like the climate of our planet, we cannot hope that the equations that we write down perfectly describe the exact behavior of a given system. However, we hope that these models still tell us information about these systems, and that is precisely the aim of Conley theory.
This theory is well developed for flows, but differential equations with discontinuous right hand sides are not as well understood. In fact, these systems generally do not have unique solutions, and so Richard McGehee introduced a set-valued analogue of the flow, the multiflow, in order to study these systems. In the papers listed below, I begin to generalize Conley Index Theory to multiflows.
Papers
These papers were both published in Topological Methods for Nonlinear Analysis in 2022, and I am the sole author on each of them:
Isolating neighborhoods and their stability for differential inclusions and Filippov systems
Conley index theory and the attractor-repeller decomposition for differential inclusions
For a more complete view of this research, check out my PhD thesis: