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 \(\varepsilon\), 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 \(\varepsilon\) does not provide a good
ansatz. We overcome these two points through a combination of
numerical analysis, matched asymptotics, and techniques from
functional analysis.
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