Impurities in Spatially Extended Pattern Forming Systems
Pattern formation can be found in many biological, chemical, and physical systems. Examples include vegetation patterns, neural networks, oscillating chemical reactions, convection cells, and many others. In this talk we study the role of impurities in shaping patterns. In particular, we concentrate on two dimensional spatially extended pattern forming systems. We look at this problem from the point of view of perturbation theory and focus on one example inspired by physical phenomena: how impurities act as pacemakers and generate target patterns in an array of oscillators. The impurity, which we represent as a localized function of strength $\eps$, can be included as a perturbation to the model equation representing our system. However, a standard argument using the implicit function theorem is not possible since the analysis presents two challenges. First, the linearization about the steady state is not invertible in regular Sobolev spaces. Second, the nonlinearities play a major role in determining the relevant approximation, so that a regular perturbation expansion in $\eps$ does not provide a good ansatz. We overcome these two points through a combination of numerical analysis, matched asymptotics, and techniques from functional analysis.