CENTER MANIFOLDS WITHOUT A PHASE SPACE
CENTER MANIFOLDS WITHOUT A PHASE SPACE
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DOI：
10.1090/tran/7190
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发表时间：
2018-08-01
影响因子：
1.3
通讯作者：
Scheel, Arnd
中科院分区：
文献类型：
--
作者：
Faye, Gregory;Scheel, Arnd
我们建立中心流形定理，使人们能够研究小的分歧解决方案，从一个平凡的状态，在系统的功能方程提出的真实的线。这类方程包括最重要的非线性方程与非局部耦合通过卷积算子，因为它们出现在神经科学的空间扩展动力学的描述。这些系统具有自然的空间平移对称性，但由于通过非局部卷积算子的无限范围的向前和向后耦合，与这种空间移位相关的空间演化的局部存在性或唯一性定理，甚至诱导动力学的相空间的动机良好的选择似乎是不可用的。我们进行减少完全依赖于功能分析方法。尽管问题的非局部性质，我们恢复一个局部微分方程描述的动态上的一组小有界的解决方案，利用原来的问题的平移不变性诱导流动行动的中心流形。我们应用我们的减少程序的问题，在数学神经科学，特别是说明了新类型的代数计算泰勒喷气机减少向量场。
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.