Nonlinear Dynamics at the Free University Berlin

Summer 2012

Seminar Advanced Topics in Nonlinear Dynamics

PD Dr. P. Gurevich, Dr. Stefan Liebscher


Apr 12, 2012 Ali Ahmadi
(FU Berlin)
Exponential Limit Shadowing Property
In this lecture, we introduce the notion of exponential limit shadowing property and show that this is a persistent property near a hyperbolic set of a dynamical system. We see that the Omega-stability implies the exponential limit shadowing property and give an example to show that this property is different from limit shadowing.
Sergey Tikhomirov
(FU Berlin)
Shadowing lemma for partially hyperbolic systems
We say that diffeomorphism $f$ of a manifold $M$ is partially hyperbolic if the tangent bundle of $M$ admits an invariant splitting $E^s + E^c + E^u$ such that $E^s$ and $E^u$ are uniformly hyperbolic and $E^c$ is not. If $E^c$ is empty, the diffeomorphism is uniformly hyperbolic. The shadowing lemma says that in hyperbolic systems any pseudotrajectory can be shadowed by an exact trajectory. We introduce notion of central pseudotrajectory and prove that in partially hyperbolic systems any pseudotrajectory can be shadowed by a central pseudotrajectory.
Apr 19, 2012 Brian Smith
(FU Berlin)
Invariant manifolds for quasilinear parabolic equations
We will investigate the existence of a strongly stable manifold of solutions for a certain class of quasilinear parabolic equations. The aim is to try to obtain such a theorem even when there is a null eigenvalue of the linearized operator involved.
Apr 26, 2012 Tuyen Vu Xuan
(FU Berlin)
Delay-Gleichungen: auf der Suche nach einem globalen Attraktor
Stefan Liebscher
(FU Berlin)
The tumbling Universe: Dynamics of Bianchi Models in the Big-Bang Limit
Bernold Fiedler
(FU Berlin)
Delayed feedback control of delay equations
May 03, 2012 N.N. t.b.a.
May 10, 2012 Alan Rendall
(FU Berlin)
Signalling pathways in T cells and chemical reaction network theory
Chemical reaction network theory (CRNT) is a collection of tools for understanding the global dynamics of solutions of the type of systems of ODE which come up in modelling chemical reactions, in particular in biology. When the tools (such as the deficiency zero theorem) can be applied they give strong results, even for systems with large numbers of unkowns and parameters. The question arises as to how often systems for which CRNT is useful come up in real applications. I will mention a few examples where this is this case, including some arising in T cell signalling. In particular I will present some work I did recently on a model due to Salazar and H\"ofer for a signalling pathway involving the transcription factor NFAT (nuclear factor of activated T cells). This is system of 56 equations with 134 parameters. It can be shown that for any fixed values of the parameters and an obvious conserved quantity this system has a unique stationary solution and that all other solutions converge to it at late times. This model describes only part of the NFAT signalling pathway and I will also describe what happens when it is coupled to a model for another part of the pathway involving calcium oscillations. If time permits I will also indicate how changing the model to explicitly include enzyme concentrations could lead to more complicated dynamics.
May 24, 2012 Isabelle Schneider
(FU Berlin)
Stabilization of symmetrically coupled oscillators by time-delayed feedback control
We consider symmetrically coupled oscillators in Hopf normal form. We focus on the case of three oscillators as the simplest example admitting nontrivial symmetry. Our aim is to stabilize the inherently unstable periodic orbits with discrete rotating wave symmetry. These orbits emerge from a symmetric Hopf-bifurcation. Time delayed feedback control (Pyragas control) is introduced. Going beyond previous results by Fiedler et al., we use the spatio-temporal symmetry of the system to establish a noninvasive control method. For that purpose we introduce symmetry-adapted coordinates. Due to a suitable control term the symmetric Hopf bifurcation splits into simple Hopf bifurcations with a two dimensional central manifold. As a consequence standard exchange of stability in two dimensions is now applicable. As a result, we are for the first time able to state explicit analytic conditions for the stabilization of three oscillators. These conditions are necessary and sufficient. We distinguish between the sub-and supercritical cases and within the subcritical case between hard and soft springs.
May 31, 2012 Martin Vaeth
(FU Berlin)
Introduction to degree theory of function triples
The aim of the talk is to give an introduction to degree theory with some outlook to recent extensions like degree theory with Fredholm maps, multivalued maps, and function triples. Some difficulties in the theory are sketched and some examples will be given.
Jun 07, 2012 Poster
Jun 14, 2012 Poster
Jun 21, 2012 Konstantin Bubolz
(FU Berlin)
Delay stabilization in coupled oscillator systems
Anna Karnauhova
(FU Berlin)
Monotone Lyapunov functions in Bianchi models
Jun 28, 2012 Mustapha Charafi
(IPP Casablanca)
Two dimensional mathematical modelling of sediment transport and morphological processes in a free surface flow
Vu Xuan Tuyen
(FU Berlin)
Kaplan's first eigenvalue method
Jul 02, 2012 Poster
Jul 12, 2012 Konstantin Mischaikow A combinatorial framework for nonlinear dynamics

Time and Place

Talks usually take place on Thursday at 2:15 p.m.
at the Free University Berlin
Room 130, Arnimallee 3 (rear building), 14195 Berlin.

Guests are always welcome!


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