Coherent Structures in Spatially Extended Systems
Transitions from simple to complicated dynamics in spatially extended systems are often organized around simple spatio-temporal building blocks. Temporal dynamics evolve around periodic, homoclinic and heteroclinic solutions. Similarly, spatial patterns in large systems are organized around wave trains and localized, coherent structures. Coherent structures include simple pulses and fronts, but also more complicated patterns such as corners in two-dimensional interfaces, target patterns, spiral waves, disclination and line defects. The lecture will explain why these structures appear in a robust fashion in a large class of dissipative systems, and how transition to turbulent dynamics is organized around such simple building blocks. The major obstacle in the analysis of coherent structures are essential spectra, which reflect the many active degrees of freedom in large systems. We will show how this difficulty can be overcome when one views coherent structures as spatially heteroclinic orbits. Robustness, bifurcation, and stability analysis then naturally separate into a local center manifold analysis (for the essential spectrum), and a global Melnikov analysis (for the point spectrum).
Arnd Scheel, University of Minnesota