期刊
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
卷 370, 期 8, 页码 5843-5885出版社
AMER MATHEMATICAL SOC
DOI: 10.1090/tran/7190
关键词
Center manifolds; nonlocal equation; Fredholm operators
类别
资金
- National Science Foundation [NSF-DMS-1311740, NSF-DMS-1612441]
- DAAD Fellowship
- ANR project NONLOCAL [ANR-14-CE25-0013]
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.
作者
我是这篇论文的作者
点击您的名字以认领此论文并将其添加到您的个人资料中。
推荐
暂无数据