Winter 2009/2010
Forschungsseminar Dynamische Systeme
(Seminar für Diplomanden und Doktoranden)
(Internes Seminar der Arbeitsgruppe Nichtlineare Dynamik)
Prof.
Dr. Bernold Fiedler
Juliette Hell,
Dr. Stefan Liebscher
Program
Oct 22, 2009 
Orkan Konc 
The Pascal Triangle as a Dynamical System 
Nov 5, 2009 
Alexander Slepukhin (St.Petersburg State University) 
Nonautonomous Global Attractors 
Nov 12, 2009 
Prof. Dr. Sergey Yu. Pilyugin (St.Petersburg State University) 
Orbital shadowing property for diffeomorphisms 
We show that the C^{1}interior of the set of diffeomorphisms
having the orbital shadowing property coincides with the set of structurally
stable diffeomorphisms.

Nov 19, 2009 
Alexander Slepukhin (St.Petersburg State University) 
Introduction to the nonintegral dimension theory 
For some sets which have complex structure (like fractals),
the usual concept of dimension may be not sufficient.
We will introduce concepts of noninteger dimensions such as fractal and Hausdorff dimensions
and consider their properties, relations between them, and methods to calculate them.
Then we will study the behaviour of the fractal and Hausdorff dimensions on spaces
which have Cartesian product structure.
It is useful because the phase space of extended dynamical systems corresponding
to nonautonomous differential equations has the Cartesian product structure.
At the end, we will apply this theory for estimating the dimension
of nonautonomous global attractors arising in the problem under consideration.

Nov 26, 2009 
Prof. Dr. Xiaojing Yang (Tsinghua University, Beijing) 
Boundedness and unboundedness of solutions of second order differential equations 
In this talk, I shall first give a general introduction to the boundedness and unboundedness problem of solutions
of second order nonlinear ordinary differential equation.
Then I give a simple description of the ideas of proving the boundedness and unboundedness problems of solutions.
Last, I formulate some open questions and future research problems.

Dec 10, 2009 
Saul Ares (MPI Physics of Complex Systems, Dresden),
Luis Morelli (University of Buenos Aires, Argentina) 
Discrete and continuum description
of nonidentical coupled phase oscillators with delayed coupling 
Spatiotemporal patterns can arise in arrays of interacting oscillators. Due
to intrinsic lags in signal production, propagation, and processing, the
interaction between oscillators is not instantaneous. In cellular systems,
for example, slow complex kinetics introduce a time delay in intercellular
coupling. Using a generic description based on coupled phased oscillators,
we study the effects of delayed coupling on patterning and timing. We derive
a continuum description for a system of coupled phase oscillators in which
the coupling between neighboring cells is arbitrarily delayed. We show that
delays introduce a renormalization of the coupling strength and autonomous
frequencies of individual oscillators. We find analytical solutions for the
phase profile and collective frequency in both discrete and continuum
descriptions, and derive the stability threshold for discrete solutions.
Motivated by a biological system, we discuss the open challenges of
introducing a spatial drift in the problem: together with the delay, this
introduces effective nonlocal coupling and exotic boundary conditions.

Dec 17, 2009 
Johannes Buchner 
Inhomogeneous cosmologies and the BKLconjecture 
n mathematical cosmology, one of the big open questions is the structure
of the inital singularity (i.e. the big bang). According to a conjecture
of Belinskii, Khalatnikov and Lifshitz (BKL) from the 1970s, the approach
is vacuum dominated, local and oscillatory (labeled "Mixmaster").
This "BKLpicture" is supported by many heuristics and numerical
calculations, but mathematical proofs exist only in very special cases,
like in spatial homogeneous spacetimes (where the Einstein Field Equations
reduce to ODEs) and even here the picture is not complete.
I will present some ideas how to tackle the problem in the spatially
inhomogeneous case with less but still some symmetry  e.g. Gowdy and
G2models that lead to 1dimensional PDEs. Furthermore, I will discuss the
question if the formation of spikes (i.e. unbounded growth of spatial
derivatives in the approach to the initial singularity) is consistent with
the BKLpicture or not.

Dec 18, 2009
1012 a.m. 
Prof. Dr. Alan Rendall (MPI Gravitationsphysik, Potsdam) 
Introduction to Black Holes 
This talk is an elementary introduction to black holes. First I review
some background from Newtonian physics. Then I discuss the Schwarzschild
solution of the vacuum Einstein equations which represents a
nonrotating, uncharged
black hole. Then I briefly describe the Kerr and ReissnerNordström
solutions which represent rotating and charged black holes,
respectively. I explain some conceptual and technical difficulties
involved in the question of the dynamical stability of these solutions.
Finally, I say something about the formation of black holes from regular
initial data.

Jan 7, 2010 
Christian Kuehn (Cornell University) 
Multiple timescale dynamics: bifurcations and mixedmode oscillations 
First, we will give an introduction to fastslow systems.
The geometric viewpoint of the theory will be emphasized. Then we
discuss the threedimensional FitzHughNagumo (FHN) equation and
its bifurcations. The singular limit bifurcation diagram of the FHN
equation will be derived. We shall also look at mixedmode oscillations
(MMOs) in the FHN equation and outline the role of MMOs in chemistry,
neuroscience and physics.

Jan 14, 2010 
Prof. Dr. Arnd Scheel (University of Minnesota) 
Coherent structures and Liesegang pattern formation 
Liesegang rings form in strikingly simple experimental setups.
The resulting patterns, sometimes regular, sometimes irregular, are ubiquitous
in nature  yet mathematically poorly understood. We present models and
results that lead to a conceptual understanding of existence and creation of such
patterns, based on existence, stability, and bifurcation results for
coherent structures in our models. A crucial role is played by pushed and pulled
fronts in spinodal decomposition scenarios.

Jan 21, 2010 
Eyal Ron 
Traveling 2D waves in the AresMorelli Model 
The Ares Morelli model is a mathematical Kuramoto based model that describes development of vertebrates. It was presented to us in a seminar talk in December 2009. After that talk, Prof. Fiedler proved existence of traveling wave solutions to a onedimention AresMorelli model. Based on Prof. Fiedler's proof, I will present traveling wave solutions to the Ares Morelli model on a 2D square lattice that supports a plane frequency profile. In addition, a 2D hexagonal lattice with radial frequency profile will be discussed, as it is a better approximation to reality.

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 !
Archive
