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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Center manifolds without a phase space

Authors: Grégory Faye and Arnd Scheel
Journal: Trans. Amer. Math. Soc. 370 (2018), 5843-5885
MSC (2010): Primary 37G99; Secondary 34B40
Published electronically: April 17, 2018
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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.

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Additional Information

Grégory Faye
Affiliation: CNRS, UMR 5219, Institut de Mathématiques de Toulouse, 31062 Toulouse Cedex, France

Arnd Scheel
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455

Keywords: Center manifolds, nonlocal equation, Fredholm operators
Received by editor(s): November 22, 2016
Received by editor(s) in revised form: January 18, 2017
Published electronically: April 17, 2018
Additional Notes: The first author was partially supported by the ANR project NONLOCAL ANR-14-CE25-0013
The second author is the corresponding author
The second author was partially supported by the National Science Foundation through grants NSF-DMS-1311740 and NSF-DMS-1612441 and through a DAAD Fellowship.
Article copyright: © Copyright 2018 American Mathematical Society