Some tools for understanding pattern formation in oscillatory media with nonlocal coupling
Oscillatory media refers to any system that is composed of small elements that oscillate and that interact with each other via some form of weak coupling. Examples include groups of fireflies, heart and neural tissue, and oscillating chemical reactions. The oscillatory behavior of these elements combined with local coupling gives rise to interesting patterns, from traveling fronts and pulses, to spiral waves and target patterns. When the coupling is nonlocal new patterns are possible. One example is given by chimera states, which correspond to the field of oscillators being split into regions of synchrony and regions of asynchrony. Because the nonlocal character of the coupling results in integro-differential equations as model equations, one is not able to use traditional tools from spatial dynamics to study these systems. In this talk we will look at the linearization of these equations about the homogenous state. We will prove that in the appropriate weighted space these operators are Fredholm and show how one can use this information to prove existence of solutions, derive normal forms, and develop finite element methods for these systems.