Target Patterns in a Two Dimensional Array of Oscillators with Nonlocal Coupling
Series: Analysis, Dynamics, and Applications Seminar
Location: Math 402
Presenter: Gabriela Jaramillo, Department of Mathematics, University of Arizona
It is well known that oscillating chemical reactions generate target patterns when an impurity is introduced. In this talk we show that a similar phenomena occurs in a two dimensional array of oscillators with nonlocal coupling. We concentrate on simple phase dynamics which we model after a viscous eikonal equation known to describe the phase modulation of traveling waves in reaction diffusion systems. One of the difficulties encountered in the analysis of our model comes from the linearization which is a convolution operator of diffusive type and it is therefore not an invertible operator in regular Sobolev spaces. In addition, regular perturbation does not provide a good ansatz as the nonlinearities play a major role in determining the first order approximations. We overcome these two points by deriving Fredholm properties for the linearization, and using matched asymptotics to arrive at a first order approximation.