Nonlinear Dynamics at the Free University Berlin

Sommer 2010

Forschungsseminar Dynamische Systeme

(Seminar für Diplomanden und Doktoranden)

(Internes Seminar der Arbeitsgruppe Nichtlineare Dynamik)

Prof. Dr. Bernold Fiedler

PD Dr. P. Gurevich, Dr. Stefan Liebscher


Apr 15, 2010 Gamal Mograby
(TU-FU Berlin)
Fractal geometry and applications
Der Inhalt des Vortrags gliedert sich in vier Themen:
  • Fraktionale Integration und Differentiation: Motivation und Definition der Riemann-Liouville Integrale. Eigenschaften wie die Halbgruppeneigenschaft in Lp-Raume. Beispiel. Motivation und Definition der Riemann-Liouville Ableitungen. Hinreichende Bedingung fur die Existenz der Riemann-Liouville Ableitung. Keine Vererbung der Halbgruppeneigenschaft, Diskussion an einem Gegenbeispiel. Beziehungen zwischen der Riemann-Liouville Integration und Differentiation. Weitere Eigenschaften. Eine geometrische Interpretation des Riemann-Liouville Integrals nach Podlubny. Eine physikalische Interpretation der Riemann-Liouville Integrale und Ableitungen nach Podlubny.
  • Lokale Fraktionale Differentiation: Motivation und Modifikation der Riemann-Liouville Ableitungen. Definition der lokalen fraktionalen Ableitung. Analogie zur Interpretation der gewohnlichen Ableitung als "lineare Approximation", anschliessend ein Beispiel. Weitere Eigeschaften.
  • Einfuhrung von zum Vortrag relevanten Begriffen aus der Fraktalen Geometrie, wie die Box Dimension. Methoden zur Berechnung der Box Dimension vom Graph einer Funktion.
  • Anwendung der lokalen fraktionalen Ableitung in der Fraktalen Geometrie.
Apr 22, 2010 Stefan Liebscher
(FU Berlin)
Bifurcation without parameters
Apr 29, 2010 Kseniya Darovskaya
(Peoples' Friendship University, Moscow)
On a spectral problem with integral conditions
We consider an ordinary differential operator with a spectral parameter and integral boundary conditions. An a priori estimate of solutions for sufficiently large values of the parameter is obtained. Also Fredholm solvability, sectorial structure of the spectrum and its discreteness are proved.
Eyal Ron
(FU Berlin)
A Dynamical Systems Approach to a Uniqueness Problem of a Nonautonomous Planar System
May 6, 2010 Stefan Liebscher
(FU Berlin)
The Tumbling Universe: Dynamics of Bianchi Models in the Big-Bang Limit
Pavel Gurevich
(FU Berlin)
Dynamics of parabolic equations with hysteresis
May 20, 2010 Julia Ehrt
(WIAS Berlin)
Cascades of heteroclinic orbits in hyperbolic balance laws
Alexey Osipov
(St. Petersburg State University)
Absense of roots of a C1-generic diffeomorphism and the structure of its centraliser
May 27, 2010 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications
Jun 9, 2010 Marek Fila
(Comenius University, Bratislava)
(Details on) Homoclinic and Heteroclinic Orbits for a Semilinear Parabolic Equation
Jun 10, 2010 Opening symposium of the Berlin Center for Studies of Complex Chemical Systems
Jun 17, 2010 Hans Ringström
(KTH Royal Institute of Technology, Stockholm)
The singularity in the case of T3-Gowdy
Due to the complexity of Einstein's equations, spacetimes satisfying symmetry conditions are often considered. The simplest spatially inhomogeneous cosmological solutions arise as a result of imposing the existence of a 2-dimensional group of isometries. The Gowdy class of spacetimes fall into this category. In the talk, a brief definition of this class will be given. However, the main focus will be on the asymptotics in the direction of the singularity; a description of the behaviour for a generic set of initial data as well as an outline of some of the arguments used to prove the results will be given.
Jun 24, 2010 Jens Rademacher
(Centrum Wiskunde & Informatica, Amsterdam)
Mechanisms of semi-strong interaction in multiscale reaction diffusion systems
In spatial multiscale reaction diffusion system where some diffusion lengths are much shorter than the rest, interfaces can form where only the components that diffuse on the short scale localize. The interaction between such interfaces is called semi-strong as it is driven by the nonlocalized components. Cases where the interface motion is of the order of the square of the short diffusion lengths (second order) have been studied over the past decade. By formal expansions and numerical studies we show that the interaction strength can also be of the same order as the short diffusion length (first order). These mechanisms are illustrated in the Schnakenberg model and investigate interaction manifolds and their stability. Taking a model independent point of view, starting only from a dichotomy in diffusion lengths, characteristic equations of motion of interfaces for first and second order semi-strong interaction can be derived. For first order pulse interaction with a single long diffusion length and under certain natural assumptions several explicit Lyapunov-functionals such as the largest interpulse distance are found. This is partly joint work with J. Ehrt and M. Wolfrum (WIAS, Berlin).
Jul 1, 2010 Sergio Oliva
(University Sao Paulo)
Reaction-diffusion equations with nonlinear boundary conditions with delays
We will present some techniques used to pose reaction-diffusion equations in negative fractional power spaces, following Dan Henry's book, which can be applied to nonlinear Neumann boundary conditions. This will then be adapted to study the same equations but with discrete delays in the reaction term. Finally, we present a Hopf bifurcation result.
Jul 15, 2010 Woei Chet Lim
(Albert Einstein Institute Potsdam)
pikes - nonlocal component of the generalized Mixmaster attractor
In general relativity, dynamics near spacelike singularities was described by chaotic Mixmaster/BKL oscillation. Spacetimes with two commuting Killing vector fields exhibit a new phenomenon, namely spikes, which are sub-horizon inhomogeneous structures whose dynamics differs from BKL dynamics.
I will present some background for Mixmaster oscillation, the generation of the explicit spike solution using the solution-generating transformation by Rendall and Weaver, and numerical simulations of spacetimes with two commuting Killing vector fields and solution-matching with the explicit spike solution. The numerical results suggest that the Mixmaster attractor should be generalized to include the spikes as a nonlocal component.
The last part of this work was done in collaboration with Lars Andersson, David Garfinkle and Frans Pretorius.

Time and Place

Talks usually take place on Thursday at 2:15 p.m.
at the Free University Berlin
Room 140, Arnimallee 7, 14195 Berlin.

Guests are always welcome !


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