Nonlinear Dynamics at the Free University Berlin

Winter 2010/2011

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


Oct 28, 2010 Carlos Rocha
(Instituto Superior Tecnico, Lisbon, Portugal)
A permutation characterization of Sturm global attractors of Hamiltonian type
We consider semiflows generated by Neumann boundary value problems of the form ut = uxx + f(u) on the interval 0 ≤ x ≤ π for dissipative nonlinearities f = f(u). In the much more general case f = f(x,u,ux) a permutation characterization for the global attractors of these semiflows is well known. Here we present a permutation characterization for the global attractors in the restrictive class of nonlinearities f = f(u). In this class the stationary solutions of the parabolic equation satisfy the second order ODE v'' + f(v) = 0 and we obtain the permutation characterization from a characterization of the set of -periodic orbits of this planar Hamiltonian system. This is based on a joint work with Bernold Fiedler and Mathias Wolfrum.
Nov 4, 2010 Anna Karnauhova
(Free University Berlin)
A model of hyperbolic space and the geodesic flow
Markus Mittnenzweig
(Free University Berlin)
Bistability of Actin - An Elastic Network Approach
Hannes Stuke
(Free University Berlin)
Variational principles of PDEs
Nov 11, 2010 Johannes Buchner
(Free University Berlin)
Inhomogeneous Mixmaster Cosmologies
The BKL conjecture suggests that the approach to the initial singularity ("big bang") is vacuum dominated, local and oscillatory. After some results have been achieved in an spatially homogeneous setting of Bianchi class A ("Mixmaster"-models) and in an inhomogeneous, but non-oscillatory setting of Gowdy-spacetimes, the next step is to consider inhomogeneous oscillatory cosmological models. The simplest interesting case is given by the G2-cosmologies.
I will present a version of the evolution equations of G2 models that directly contain the the Gowdy spacetimes as well as the oscillatory Bianchi-model of class B. Then I will focus on the latter and explain the "frame transitions" that appear in the oscillations towards the singularity. Finally, I show the existence of a heteroclinic 3-cycle and discuss its properties in the spirit of results that have been achieved in Bianchi class A.
A particular complication in inhomogeneous models is the occurrence of spikes, i.e. the formation of spatial structure. In an outlook, I will discuss how the knowledge of the spatially homogeneous background dynamics helps to understand the formation of true and false spikes in G2 models.
Nov 25, 2010 Johannes Buchner
(Free University Berlin)
Partially Hyperbolic Fixed Points and Cosmology
In the paper "Partially Hyperbolic Fixed Points" by Floris Takens, published in "Topology Vol. 10", 1971, a local linearization theorem is proved for diffeomorphisms and fixed points of flows that are not hyperbolic but satisfy certain "Sternberg Non-Resonance Conditions". In my talk, I will review the result and try to give an overview of the proof. If time permits, I will also discuss the application of the result to cosmological models of Bianchi type IX (as done by Francois Béguin) as well as possible future applications of the result to Bianchi models of class B.
Dec 2, 2010 Taras Girnyk
(WIAS Berlin)
Multistability of twisted states in non-locally coupled Kuramoto-type models
We consider a ring of N Kuramoto oscilators with non-local repulsive coupling. We obtain sufficient conditions for stability of so-called twisted states. We discover new types of solutions and establish a basic framework for their description. We discover phenomenon of spatial chaos in repulsive networks of Kuramoto oscillators.
Dec 9, 2010 Hannes Stuke
(Free University Berlin)
A variational principle for PDEs
It is well known that a weak solution of certain evolution PDE could be obtained from the minimizer of some energy functional. In this talk, I will present a variational principle mostly developed by Nassif Ghoussoub, which allows the variational resolution for many non-Euler-Lagrange type PDEs.
Dec 16, 2010 Juliette Hell
(Free University Berlin)
News of the BKL conjecture
I will try to give an overview of the paper by Reiterer and Trubowitz, "The BKL Conjectures for Spatially Homogeneous Spacetimes". They prove tumbling universe (for Bianchi VIII and IX, vacuum) for a more "generic" class than the other results so far (Beguin, Liebscher et al): they exhibit a set of initial conditions with positive Lebesgue measure showing this behaviour.
Jan 13, 2011 Martin Vaeth
(Free University Berlin)
Bifurcation for a Reaction-Diffusion System with Pure Neuman Boundary Conditions and Inequalities
For a reaction-diffusion system which is subject to Turing's effect of diffusion-driven instability, endowed with Neumann-Signorini boundary conditions, there is a branch of bifurcations of stationary solutions in a parameter domain where the same system with only classical Neumann conditions is stable. In the presence of additional Dirichlet conditions, this is well known, but for pure Neumann boundary conditions, such a bifurcation was an open problem for a long time. The main purpose of the talk is to sketch the proofs, in particular how to calculate the Leray-Schauder degree for the maps associated to the problem.
Jan 20, 2011 Alexander Skubachevskii
(Peoples' Friendship University of Russia)
The Vlasov-Poisson Equations in Half-Space
The Vlasov-Poisson equations describe the evolution of densities for electrons and ions in rarefied plasma. Existence and uniqueness of generalized solutions for such equations in domains with boundary or in the whole three-dimensional space was studied by many mathematicians. In this lecture we consider existence and uniqueness of classical solutions to the Vlasov-Poisson equations in half-space.
Jan 27, 2011 Johannes Buchner
(Free University Berlin)
The Proof of Taken's Linearization Theorem
Taken's linearization theorem provides a local linearization for fixed points of diffeomorphisms or flows that are not hyperbolic but satisfy certain Sternberg Non-Resonance Conditions. It is a generalization of the theorems of Grobman-Hartman and Sternberg, which provide a linearization near a hyperbolic fixed point. I will present the proof and focus on the case that is relevant for the application of the theorem to Bianchi cosmologies, and restrict to that case (line of equilibria, diagonal linearization, etc) where helpful.
Feb 3, 2011 Johannes Buchner
(Free University Berlin)
The Proof of Taken's Linearization Theorem, Part II

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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