Applied & Computational Mathematics seminar: Pattern-forming fronts in the FitzHugh–Nagumo system

Speaker: Prof. Paul Carter, University of California, Irvine

Title:  Pattern-forming fronts in the FitzHugh–Nagumo system

Abstract:  When a spatially homogeneous state destabilizes, localized perturbations can grow into large amplitude spatial patterns, which spread into the bulk, invading the unstable state. The nature and properties of the patterns which appear, such as wavelength, orientation, and amplitude, are frequently determined by the behavior in the leading edge of the spreading process. We consider such pattern-forming fronts in the FitzHugh–Nagumo PDE in the so-called oscillatory regime. The pattern is selected from a family of periodic traveling wave train solutions by an invasion front. Using geometric singular perturbation techniques, we construct “pushed” and “pulled” pattern-forming fronts as heteroclinic orbits between the unstable steady state and a periodic orbit representing the wave train in the wake. In the case of pushed fronts, the wave train necessarily passes near a pair of nonhyperbolic fold points on the associated critical manifold. We also discuss implications for the stability of the pattern-forming fronts and the challenges introduced by the fold points in the corresponding spectral stability problem.

Time:  Friday, April 11 – 1:30pm – 2:30pm

Place:  Exploratory Hall, room 4106