Continuation Sheaves in Dynamics: Sheaf Cohomology and Bifurcation

TL;DR

This paper introduces a categorical sheaf-theoretic framework for analyzing how algebraic structures like attractors and invariant sets in dynamical systems evolve with parameters, using sheaf cohomology to detect bifurcations.

Contribution

It develops a novel sheaf-theoretic approach to dynamical systems, linking algebraic invariants to bifurcation analysis through category theory and cohomology.

Findings

Sheaves encode continuation of algebraic structures in dynamical systems.

Sheaf cohomology provides invariants that detect bifurcations.

Framework applies to lattice and ring structures in dynamics.

Abstract

Continuation of algebraic structures in families of dynamical systems is described using category theory, sheaves, and lattice algebras. Well-known concepts in dynamics, such as attractors or invariant sets, are formulated as functors on appropriate categories of dynamical systems mapping to categories of lattices, posets, rings or abelian groups. Sheaves are constructed from such functors, which encode data about the continuation of structure as system parameters vary. Similarly, morphisms for the sheaves in question arise from natural transformations. This framework is applied to a variety of lattice algebras and ring structures associated to dynamical systems, whose algebraic properties carry over to their respective sheaves. Furthermore, the cohomology of these sheaves are algebraic invariants which contain information about bifurcations of the parametrized systems.