Nonlinear Dynamics at the Free University Berlin

Winter 2008/2009

Seminar für Diplomanden und Doktoranden

(Internes Seminar der Arbeitsgruppe Nichtlineare Dynamik)

Prof. Dr. Bernold Fiedler

Dr. Marc Georgi, Dr. Stefan Liebscher


Program

Oct 16, 2008 Almut Burchard
(University of Toronto)
Motion of an infinite elastic curve: well-posedness, stability of solitary waves, open questions
Oct 23, 2008 Alexey Teplinsky
(Academy of Sciences, Kiev)
Herman's theory revisited
For certain range of r and δ we prove that any Cr-smooth circle diffeomorphism with irrational rotation number from the Diophantine class Dδ is Cr-1-δ-smooth conjugate to a rigid rotation. This is the only sharp result of this kind for today. In particular, it improves the famous results of M.Herman and his followers. The proof is tricky but almost elementary. (Joint work with K.Khanin.)
Oct 30, 2008 Sergey Tikhomirov Shadowing in smooth flows
We study the shadowing property of smooth vector fields on a compact manifold. The shadowing problem is related to the following question: under which condition for any pseudotrajectory of a flow generated by vector field there exists a close exact trajectory? The main difference between the shadowing problem for vector fields and the similar problem for discrete dynamical systems is related to the necessity of reparametrization of shadowing trajectories in the former case.
In this work we study the structure of the C1-interiors of the sets of vector fields with various shadowing properties. We show that for the case of Lipschitz shadowing property such a C1-interior coincides with the set of structurally stable vector fields. If the dimension of the manifold is less or equal 3, then the similar statement can be proved for the Oriented shadowing property.
The main result I present in this talk is an example of not structurally stable vector field on 4-dimensional manifold that lies in the interior of the set vector fields with Oriented shadowing property.
Nov 06, 2008 Marc Georgi
(Freie Universität Berlin)
Travelling waves in the Fermi-Pasta-Ulam lattice
In the talk I will motivate the problem and mainly focus on the main ingredient in the proof, which is Lin's method.
Nov 13, 2008 Bernold Fiedler
(Freie Universität Berlin)
Aus nichts wird nichts ?
Nov 27, 2008 Marc Georgi
(Freie Universität Berlin)
Lin's method
In the talk I will introduce a functional analytic frame- work, with which one can detect periodic and multi-round homoclinic orbits near a primary homoclinic orbit. I will mainly focus on ordinary differential equations but also point out technical difficulties when it comes to advance delay equations.
Dec 04, 2008 Hao Wu
(WIAS Berlin)
Convergence to Equilibrium for Nonlinear Evolution Equations
In this talk I will present the Lojasiewicz-Simon approach on the study of convergence to equilibrium for nonlinear evolution equations. Some applications of this approach will be discussed.
Dec 11, 2008 Bernahrd Brehm
(Freie Universität Berlin)
Invariant measures of the Kasner map
Jan 08, 2009 Bernd Kawohl
Universität zu Köln
Variational versus PDE-based Approaches in Mathematical Image Processing
In image processing we are often presented with amazing examples of image enhancement algorithms. Yet, when applied to different noisy images, they can product unwanted effects. The analysis of such algorithms lags behind their intuitive development. Two essentially different models have found wide recognition: a variational approach according to Mumford and Shah and an approach via nonlinear diffusion equations. One of these equations is nonparabolic and was suggested by Perona and Malik. In this short survey I point out a connection between these two seemingly unrelated approaches and explain some connection with total variational flow.
followed by the colloquium talk
5:15 p.m., Lecture hall 001, Arnimallee 3
Gleichdicke, oder warum konvexe Geometrie Leben retten kann
Wann hat ein Rohr einen exakt kreisförmigen Querschnitt? Wenn es aus jeder Richtung von außen gleich dick zu sein scheint? Das könnte man mit einer Schieblehre nachmessen, und so gaben es die Vorschriften der NASA beim Raketenbau vor. Den Verfassern der Vorschriften war entgangen, dass es geometrische Formen gibt, so genannte "Gleichdicke", die nicht Kreise sind. Letztlich führte dieser Irrtum zum Tod einer ganzen Raumschiffbesatzung. Im Vortrag wird dargelegt, wo uns Gleichdicke im täglichen Leben überall begegnen und welche interessanten mathematischen Fragen mit ihnen verbunden sind. Der Vortrag wurde zum Jahr der Mathematik für die mathematisch interessierte Öffentlichkeit entworfen und richtet sich daher auch an Hörer aller Semester und anderer F├Ącher.
Jan 15, 2009 Mihaela Enculescu & Juliane Zimmermann
(Max Delbrück Center for Molecular Medicine)
Modelling actin-based cellular motility
Cell crawling is a vital process, playing an essential role in embryonic development, wound healing and immune response, but is also responsible for cancer metastasis. The driving phenomenon of cell motility is the protrusion of the cell's plasma membrane and the formation of a thin flat structure at the leading edge of the cell, called the lamellipodium, by means of actin filaments. Actin is a highly dynamic biopolymer, that steadily binds monomers at one end and looses monomers at the other end. We model the formation of a lamellipodium by calculating the force which the semiflexible actin filaments exert on the membrane. This force is crucially dependent on the length of the fluctuating filaments, which are cross-linked at the inner parts of the lamellipodium thereby forming an "actin gel". When analyzing the dynamics of the filament's lengths, we obtain stationary and oscillatory movements. We also solved hydrodynamic equations to include the velocity of the actin gel into our model. As we regard the lateral extension of the lamellipodium, we find the spreading of different wave forms along the plasma membrane. Those morphodynamic patterns are in good agreement with experimental results.
Jan 22, 2009 Stefan Liebscher
(Freie Universität Berlin)
Heteroclinic cycles in food-chain models
Jan 29, 2009 Julia Ehrt
(WIAS Berlin)
Attractors in hyperbolic equations and parabolic regularizations
Feb 05, 2009 Abderrahim Azouani
(Freie Universität Berlin)
P(x)-Laplacian
Feb 12, 2009 Seminar Unendlichdimensionale Dynamik

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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