Oberseminar Nonlinear Dynamics

Diffusively coupled dynamical systems with arbitrary coupling graphs are considered. Explicit upper bounds for the minimal coupling strength (diffusion constant) needed to achieve complete synchronization are derived using Lyapunov functions in the difference variables. These bounds are a product of a term depending only on the individual system dynamics and a term depending only on the coupling graph. The latter is formulated purely in graph theoretical terms. Furthermore, systems where all or part of the couplings are switched on and off in a random fashion are considered. It is proved that for sufficiently fast switching, complete synchronization is almost always achieved, if the averaged system completely synchronizes. This is applied to small-world networks where in addition to fixed couplings "short-cut" couplings are switched on and off. It is shown that even with very low probability of switching short-cuts on, the coupling strength needed to achieve complete synchronization can be considerably lowered.

We study existence and stability of curved interfaces in planar reaction-diffusion systems. We start with a planar, x-independent front and investigate the existence of traveling waves which locally resemble the primary front. The problem of existence of corners is reduced to an ordinary differential equation that can be viewed as the travelling-wave equation to a viscous conservation law or variants of the Kuramoto-Sivashinsky equation. The corner typically but not always points in the direction opposite to the direction of propagation. For the existence and stability problem, we rely on a spatial dynamics formulation with an appropriate equivariant parameterization for relative equilibria. We also comment on oscillatory front propagation, invasion of patterns, and viscous shock waves in anisotropic systems.

14:00  Reiner Lauterbach (Universität Hamburg)

Symbolic computations in equivariant problems

14:45  Bernold Fiedler (Freie Universität Berlin)

Rotating spirals of curvature flows: a center manifold example

15:30  Kaffeepause

16:00  Jan Sieber (University of Bristol)

Dynamics of piecewise smooth delay equations

16:45  N.N. Nefedov (Staatl. Lomonossov-Universität Moskau)

Generation and propagation of fronts in IVP with periodic nonlinearity

New results on homoclinic orbits near certain generic codimension-1 and -2 heteroclinic cycles between an equilibrium and a periodic orbit are presented. In the codimension-2 case the global topology of heteroclinic sets determines the number of curves of homoclinic orbits that bifurcate and influences the leading order expansion of parameter curves. The codimension-1 case partially explains the phenomenon of 'tracefiring' in reaction-diffusion equations, which is the bifurcation of a stable excitation pulse to a self-replicating pulse-chain. For the Oregonator model the codimension-2 case can be used to understand the loss of stability of the primary pulse.