Nonlinear Dynamics at the Free University Berlin

Summer 2014

Oberseminar Nonlinear Dynamics



April 22, 2014 CANCELLED
April 29, 2014
4:15 p.m.
Boris Hasselblatt
(Tufts University, Medford, MA/US)
Statistical properties of deterministic systems by elementary means
The Maxwell-Boltzmann ergodic hypothesis aimed to lay a foundation under statistical mechanics, which is at a microscopic scale a deterministic system. Similar complexity was discovered by Poincaré in celestial mechanics and by Hadamard in the motion of a free particle in a negatively curved space. We start with a guided tour of the history of the subject from various perspectives and then discuss the central mechanism that produces pseudorandom behavior in these deterministic systems, the Hopf argument. It has been known to extend well beyond the scope of its initial application in 1939, and we show that it also leads to much stronger conclusions: Not only do time averages of observables coincide with space averages (which was the purpose for making the ergodic hypothesis), but any finite number of observables will become decorrelated with time. That is, the Hopf argument does not only yield ergodicity but mixing, and often mixing of all orders.
Please note:
Talk will take place at Konrad-Zuse-Zentrum für Informationstechnik (ZIB), seminar room, Takustr. 7, 14195 Berlin-Dahlem.
Tea/coffee will be served at 3:45 p.m. at Konrad-Zuse-Zentrum für Informationstechnik (ZIB), entrance hall, Takustr. 7, 14195 Berlin-Dahlem.
May 6, 2014 SFB 647
May 13, 2014 Marcus Hauser
(Otto-von-Guericke-Universität Magdeburg)
Interaction of Scroll Waves
Scroll waves are the three-dimensional counterparts of spiral waves occurring in excitable reaction-diffusion systems. Single scroll waves may undergo various instabilities that play an important role in the formation of cardiac arrhythmias, like ventricular tachycardia and fibrillation. While a substantial effort has been devoted to the study of the dynamics of single scroll waves, experimental investigations of the interaction of scroll waves are rare.

The interaction of a pair of scroll waves with each other was investigated. The scroll waves were created in a Belousov-Zhabotinsky reaction medium and observed by optical tomography. We studied scroll waves whose filaments were parallel to each other and oriented along the vertical axis of the reaction cylinder [1]. The dynamics of such a pair of scroll waves was found to depend on the distance between the filaments: When the distance d between the two filaments was shorter than the wavelength λ of the scroll wave (but larger than the extension of a spiral core), then the filaments displaced each other, until the inter-filament distance d became comparable to the wavelength (i.e. until d ≈ λ). Once d ≈ λ, the scroll waves rotated without being disturbed by each other.

When the distance between the two filaments was shorter than the radius of the spiral core, then two behaviours were observed: Locally co-rotating scroll waves (i.e. scrolls that presented the same sense of rotation) always repelled each other. By contrast, locally counter-rotating scroll waves may suffer a “crossover collision” [2], leading to a rupture and a subsequent reconnection of the filaments. Each of the reconnected filaments consisted of parts that originated from the two original filaments. The conditions for rupture and reconnection of filaments will be discussed.

[1] D. Kupitz, M. J. B. Hauser, J. Phys. Chem. A 117, 12711 (2013).
[2] B. Fiedler and R. M. Mantel, Documenta Math., 5, 695 (2000).
May 20, 2014 CANCELLED
May 27 Dmitry Turaev
(Imperial College London, UK)
Exponential and super-exponential growth of the number of periodic orbits in iterated function systems
June 3, 2014 SFB 647
June 10, 2014 CANCELLED
June 17 Jens Rademacher
(Universität Bremen)
Patterns in Landau-Lifschitz-Gilbert-Slonczewski equations for spintronic devices with aligned fields
The self-organized emergence of spatio-temporal patterns is a ubiquitous phenomenon in nonlinear processes on large homogeneous domains. In this talk a class of Landau-Lifshitz-Gilbert-Slonczewski equations is studied from this viewpoint, highlighting various aspects of the theory. The model describes magnetization dynamics in the presence of an applied field and a spin polarized current. Here we consider the case of axial symmetry and focus on coherent structure solutions that occur due to the symmetry in one space dimension. This is joint work with Christof Melcher (RWTH).
June 24, 2014 CANCELLED ( → SFB 910 review)
July 1, 2014 CANCELLED ( → SFB 910 review)
July 8, 2014 CANCELLED
July 16
Hiroshi Matano
(University of Tokyo)
Propagating terrace for semilinear diffusion equations
In this talk I will discuss the behavior of spreading fronts in semilinear diffusion equations on the entire space. Here, by a "spreading front", I mean the expanding level sets of a solution that starts from compactly supported (or rapidly decaying) nonnegative initial data.
If the nonlinearity is multi-stable, the dynamics of a solution can no longer be described by a single front, but by what we call a "propagating terrace", which roughly means a layer of several fronts that expand to infinity at different speeds.
I will first give a brief review of my ealier result for the one-dimensional case (joint work with Thomas Giletti and Arnaud Ducrot), in which the term "propagating terrace" was first introduced. I will then discuss the higher dimensional case, and show that every solution eventually behaves like what we call a "radially symmetric propagating terrance". This latter part is joint work with Yihong Du.

Please note:
Talk will take place at WIAS Berlin (Langenbach-Seminar), Ehrhard-Schmid-Hörsaal, Mohrenstr. 39, 10117 Berlin.

Time and Place

Talks usually take place on Tuesday at 3:15 p.m.
at the Free University Berlin
Arnimallee 3 (rear building), room 130.

Tea/coffee at 2:45 p.m.
Arnimallee 3 (front building), room 006.
Guests are always welcome !


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