PDE Seminar: "Invasion pattern fronts near the edge of the homoclinic snake", David Lloyd (University of Surrey)
Kontakt: erik [dot] wahlen [at] math [dot] lu [dot] se
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For the past two decades, extensive research has been looking at the formation of stationary localised patterns of reaction-diffusion PDEs near a subcritical Turing instability. The typical bifurcation scenario (known as “homoclinic snaking”) involves an unstable small amplitude localised pattern (of the rough form sech(x)cos(x)) bifurcating at the Turing bifurcation, that then undergoes an infinite sequence of folds as a bifurcation parameter is varied, where the localised pattern alternates in between being stable and unstable and develops a periodic core. Despite much progress in understanding the mathematical structure of homoclinic snaking, little is known once one steps outside the this region other than the patterns should start to move. In this talk, I will give an overview of what happens and present a novel numerical algorithm based on an analysis technique to explore patterns outside the homoclinic snaking region.