Nonlinear Dynamics at the Free University Berlin

Sommersemester 2004

Oberseminar Nonlinear Dynamics




Apr. 13th 2004 Prof. Dr. Martin Hasler
Swiss Fed. Inst. of Technology, Lausanne
Complete synchronization of regular and small-world networks of dynamical systems

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.

Apr. 20th 2004

3:45 p.m.
Prof. Dr. Arnd Scheel
University of Minnesota, Minneapolis
Corner defects in interface propagation

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.

Apr. 27th 2004

2:00 p.m.
Festkolloquium zu Ehren von Klaus Schneider
WIAS Berlin, 10117 Berlin, Mohrenstr. 39, Erhard-Schmidt-Hörsaal
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
May 11th 2004 Dr. Dmitrii Rachinskii
Sector estimates in Hopf bifurcation problems
May 18th 2004

2:30 p.m.
Prof. Dr. Leonid Cherkas
Minsk, State University
Limit cycles of polynomial vector fields
WIAS Berlin, 10117 Berlin, Mohrenstr. 39, Erhard-Schmidt-Hörsaal
May 25th 2004 Prof. Dr. Dan Luss
University of Houston
Temperature patterns in catalytic reactors
June 8th 2004 Dr. Jens Rademacher
University of Minnesota
Homoclinic orbits near heteroclinic cycles with periodic orbits

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.

June 22nd 2004 Dr. Sarah Day
Cornell University
Conley index techniques for global dynamics: a study of the Swift-Hohenberg equation
June 29nd 2004 Prof. Mohamed Belhaq
Univ. Hassan II, Casablanca
Fast Vibrations in Mechanical Systems
July 6th 2004 Prof. Dmitry Turaev
University of Ber Sheva, Israel
On maps close to identity

Time and Place

Talks usually take place on Tuesday at 3:15 p.m.
at the Free University Berlin, Department of Mathematics/Computer science
in room 108/109, Arnimallee 6 (the "Pi-Building"), 14195 Berlin.

Tea and coffee will be served at 2:45 p.m. on the ground floor.
Guests are always welcome !


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