Center manifolds without a phase space

TL;DR

This paper develops center manifold theorems for systems of functional equations with nonlocal coupling, enabling bifurcation analysis without relying on traditional phase space constructions, with applications in neuroscience.

Contribution

It introduces a novel reduction method for nonlocal functional equations that bypasses the need for a phase space, expanding bifurcation analysis tools for spatially extended systems.

Findings

Established center manifold theorems for nonlocal equations

Reduced dynamics to local differential equations despite nonlocality

Applied methods to neuroscience models with convolution operators

Abstract

We establish center manifold theorems that allow one to study the bifurcation of small solutions from a trivial state in systems of functional equations posed on the real line. The class of equations includes most importantly nonlinear equations with nonlocal coupling through convolution operators as they arise in the description of spatially extended dynamics in neuroscience. These systems possess a natural spatial translation symmetry but local existence or uniqueness theorems for a spatial evolution associated with this spatial shift or even a well motivated choice of phase space for the induced dynamics do not seem to be available, due to the infinite range forward- and backward-coupling through nonlocal convolution operators. We perform a reduction relying entirely on functional analytic methods. Despite the nonlocal nature of the problem, we do recover a local differential equation describing the dynamics on the set of small bounded solutions, exploiting that the translation invariance of the original problem induces a flow action on the center manifold. We apply our reduction procedure to problems in mathematical neuroscience, illustrating in particular the new type of algebra necessary for the computation of Taylor jets of reduced vector fields.