Equivariant Bifurcation from Relative Equilibria via Isomorphic Vector Fields

TL;DR

This paper develops a categorical framework for analyzing the dynamics of equivariant vector fields near relative equilibria, addressing challenges like non-smooth orbit spaces and lack of linearization, and applies it to bifurcation analysis.

Contribution

It introduces the concept of isomorphic equivariant vector fields and demonstrates their use in studying bifurcations from relative equilibria, advancing the understanding of symmetry-influenced dynamics.

Findings

Category of equivariant vector fields is equivalent to those on the slice representation.

Framework captures symmetries without passing to orbit space.

Results on genericity conditions for equivariant bifurcations.

Abstract

We present a framework for studying the dynamics of equivariant vector fields near relative equilibria. To overcome the lack of linearization at a relative equilibrium or the possible non-smoothness of the orbit space, we categorify the space of equivariant vector fields. A category where the objects are equivariant vector fields was first introduced by Hepworth in the context of smooth stacks [Hepworth, Theory Appl. Categ. 22 (2009), 542-587]. Central to our approach is the ensuing notion of isomorphic equivariant vector fields. The idea is that considering equivariant vector fields, and their corresponding dynamics, up to isomorphism is a way to take into account the symmetries of the group action without passing to the orbit space. In particular, the category of equivariant vector fields near a relative equilibrium is equivalent to the category of equivariant vector fields on the slice representation of the relative equilibrium. In this paper we apply this to bifurcations to and from relative equilibria, and to the genericity of conditions for equivariant bifurcation from relative equilibria.