Composition of Attractor Lattices

Abstract

The concept of an attractor is natural in the study of dynamical systems because it represents a behavior of the system that is likely to be directly observed in experiment or numerical simulation. An attractor and its dual repeller represent a fundamental order present in the dynamics. This order allows the algebraic theories of posets and lattices to be applied to the analysis of global dynamics. In my talk, I will discuss how the attractor lattice in composite systems can be characterized algebraically in terms of the attractor lattices of the component systems in both decoupled and coupled systems. I will begin with the decoupled case, showing both successes and limitations. I will then introduce the coupled setting, and propose a sheaf-theoretic approach to describe attractor composition. This perspective not only clarifies existing behavior but opens paths toward generalizing to systems with static inputs or feedback.