The Effects of Nonlocal Coupling in Oscillatory Media

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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.