The Effects of Nonlocal Coupling in Oscillatory Media

振荡介质这个术语是用来描述一类由表现出振荡行为的小元素组成的系统，这些小元素通过某种形式的耦合或传输机制相互连接。许多物理和生物系统都属于这一类，振荡化学反应是典型的。在这种情况下，例如，当溶液混合良好时，溶液会在两种颜色之间振荡，但当扩散发生时，就会形成有趣的图案，例如目标波和螺旋波。这种模式出现在生物、化学和材料科学的许多其他系统中，并应用于医学和技术。大多数关于振荡介质中模式形成的已知假设元素通过扩散过程在局部交流，就像在振荡化学反应中一样。但最近的实验和分析表明，非局部耦合可以产生先前未知的状态。例如，具有非局部扩散过程的化学反应可以表现出同步区域与完全不相干区域混合的结构特征。这些结构被称为嵌合体状态，也在其他具有局部、非局部和全局耦合的系统中被发现。虽然它们的存在性和稳定性已经在相振器阵列的情况下得到了证实，但对于耦合形式和所产生的模式之间的关系还不太清楚。这个项目的目的是构建一个通用的数学框架，将耦合的属性与这些和其他新模式的出现和控制联系起来。本科生参与该项目的研究。本项目重点研究具有有效非局部扩散过程的振荡化学反应，该过程可能是其中一个变量以更快的速度进化的结果。为了解决由于非局部耦合而产生的新模式，研究了螺旋嵌合体和局部目标模式的存在性，并研究了产生这些状态的条件。为了解决模式的控制，非局部反馈对螺旋核心的影响也进行了探讨。为了实现这些目标，研究者开发了卷积算子的数学理论，该理论将其傅立叶符号的形状与耦合的性质（如耦合强度和半径）联系起来。由于当参数变化时，模式会从已知的稳态中分叉，因此从摄动理论的角度探讨了模式的存在性。这需要证明稳态相关方程的线性化是Fredholm算子。也就是说，当它们被视为代数加权Sobolev空间之间的算子时，它们具有封闭范围和有限维核和核。本科生参与该项目的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The term oscillatory media is used to describe a large class of systems consisting of small elements that exhibit oscillatory behavior and which are connected to each other via some form of coupling or a transport mechanism. Many physical and biological systems fall into this category, with oscillating chemical reactions being prototypical. In this case, one has for instance a solution that oscillates between two colors when it is well mixed, but when diffusion occurs interesting patterns, such as target and spiral waves, form. Such patterns arise in many other systems in biology, chemistry, and materials sciences, with applications in medicine and technology. Most of what is known about pattern formation in oscillatory media assumes that elements communicate locally via a diffusion process, as in an oscillating chemical reaction. But recent experiments and analysis show that nonlocal coupling can give rise to previously unknown states. For example, chemical reactions with a nonlocal diffusion process can exhibit structures characterized by regions of synchrony mixed with regions of complete incoherence. These structures, called chimera states, also have been found in other systems with local, nonlocal, and global coupling. Although their existence and stability have been established in the case of arrays of phase oscillators, not much is known about the relationship between the form of coupling and the resulting pattern. The aim of this project is to construct a general mathematical framework that relates properties of the coupling to the emergence and control of these and other novel patterns. Undergraduate students participate in the research of the project.This project focuses on oscillating chemical reactions with an effective nonlocal diffusion process, which can be the result of one of the variables evolving at a faster rate. To address the emergence of new patterns due to nonlocal coupling, the existence of spiral chimeras and localized target patterns is investigated and conditions that give rise to these states are studied. To address control of patterns, the effects of nonlocal feedback on a spiral's core are also explored. To achieve these objectives, the investigator develops a mathematical theory for convolution operators that relates the shape of their Fourier symbol to properties of the coupling, such as coupling strength and radius. Because patterns bifurcate from a known steady state when a parameter is varied, the existence of patterns is approached from the point of view of perturbation theory. This requires showing that the linearizations of the relevant equations about the steady state are Fredholm operators. That is, they have closed range and a finite-dimensional kernel and cokernel when viewed as operators between algebraically weighted Sobolev spaces. Undergraduate students participate in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.