Localized States Near Supercritical Turing Bifurcations in the 1:1 Forced Complex Ginzburg-Landau Equation

Monday, June 3, 2013 - 3:30pm - 4:00pm
Keller 3-180
Yiping Ma (Northwestern University)
Much previous work on localized states has focused on the vicinity of subcritical Turing bifurcations, which create spatially uniform equilibria and spatially periodic patterns that are both stable. Indeed supercritical Turing bifurcations create stable Turing patterns together with *unstable* equilibria, which are insufficient by themselves to form robust localized patterns. However, recently robust localized patterns are found after a supercritical Turing bifurcation in the 1:1 forced complex Ginzburg-Landau equation, in a parameter regime where the unstable equilibria connect to *stable* equilibria on an S-shaped bifurcation curve. Before the Turing bifurcation, there exist localized states that resemble bounded fronts between two stable equilibria. The bifurcation structure, spectral stability, and temporal dynamics of these localized states in both one and two dimensions are determined using numerical continuation and direct numerical simulation. In particular, the bifurcation structure of 1D steady localized patterns takes the form of defect-mediated snaking that differs sharply from standard homoclinic snaking. In 2D there exist both radially symmetric localized states including circular fronts and localized rings, and fully 2D localized states including localized hexagons bounded by circular and planar fronts.

This is joint work with E. Knobloch.
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