Oberseminar Nonlinear Dynamics
|April 22, 2014
|April 29, 2014
(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.
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
|May 13, 2014
|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 . 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” , 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.
 D. Kupitz, M. J. B. Hauser, J. Phys. Chem. A 117, 12711 (2013).
 B. Fiedler and R. M. Mantel, Documenta Math., 5, 695 (2000).
|May 20, 2014
(Imperial College London, UK)
|Exponential and super-exponential growth of the number of periodic orbits in iterated function systems
|June 3, 2014
|June 10, 2014
|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
(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.
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 !