Nonlinear Dynamics at the Free University Berlin

Winter 2009/2010

Oberseminar Nonlinear Dynamics



Nov 3, 2009 Prof. Dr. Sergei Pilyugin
(St.Petersburg University)
Structural stability of diffeomorphisms and shadowing properties
We will discuss several new results concerning relations between structural stability, Ω-stability, and some shadowing properties of diffeomorphisms of smooth closed manifolds.
  1. Variational shadowing property is equivalent to structural stability.
  2. Lipschitz shadowing property is equivalent to structural stability.
  3. Lipschitz periodic shadowing property is equivalent to Ω-stability.
Some of these results are joint with A. Osipov and S. Tikhomirov.
Nov 10, 2009 Dr. Laurette Tuckermann
Turbulent-laminar patterns in plane Couette flow
The greatest mystery in fluid dynamics is probably transition to turbulence. The simplest shear flow, plane Couette flow -- the flow between parallel plates moving at different velocities -- is linearly stable for all Reynolds numbers, but nevertheless undergoes sudden transition to 3D turbulence at Re near 325. At just these Reynolds numbers, it was recently discovered experimentally at CEA-Saclay that the flow takes the form of a steady and regular pattern of wide oblique turbulent and laminar bands.
We have been able to reproduce these remarkable flows in numerical simulations of the Navier-Stokes equations. Simulations display a rich variety of variants of these patterns, including spatio-temporal intermittency, branching and travelling states, and localized states analogous to spots. Quantitative analysis of the Reynolds-averaged equations reveals that both the mean flow and the turbulent force are centrosymmetric and can be described by only three trigonometric functions, leading to a model of 6 ODEs. We find that the transition is best described as a bifurcation in the probability distribution function of the power spectrum.
SFB 647
Nov 17, 2009 Julia Ehrt
(WIAS / FU Berlin)
Promotionsvortrag / Dissertation Defense
Mathmatics of Cell Motility
15:15, Free Unversity Berlin, Room 140, Arnimallee 7 (rear building), 14195 Berlin
The talk will investigate mathematical and modelling aspects of cell motility. After giving a brief overview on the bio-chemical processes of cell motility in general I will focus on crawling cells. The different processes involved in crawling cells have posed challenging problems in terms of their physical and chemical background. In addition crawling cells pose a rich field in terms of mathematical modelling. Several aspects of the physical and chemical background and the difficulties in modeling these will be adressed in the talk. In the last part I will present a very recent model investigated by Gholami, Falcke and Frey (2008) in which interesting mathematical questions arise. I will explain the model and investigate its mathematical background.
Nov 24, 2009 Dr. Nikolay Zhuravlev
(Peoples' Friendship University of Russia)
Methoden zur qualitativen Untersuchung periodischer Lösungen von Funktionaldifferentialgleichungen
Die Eigenwerte des Monodromie-Operators beschreiben das Verhalten der Trajektorien in der Naehe einer periodischen Loesung. Im Gegensatz zu gewoehnlichen Differentialgleichungen fuehren Funktionaldifferentialgleichungen jedoch auf Operatoren zwischen unendlich-dimensionalen Raeumen. Es ist daher viel schwieriger, die Eigenwerte des Monodromie-Operators zu bestimmen. Im Vortrag werde ich ueber Methoden zur Bestimmung dieser Eigenwerte berichten.
Dec 1, 2009 SFB 647
Dec 8, 2009 Dr. Georgy Kitavtsev
(WIAS Berlin)
Reduced ODE models describing coarsening dynamics of slipping droplets
In this talk the topic of reduced ODE models corresponding to a family of one-dimensional lubrication equations derived by M"unch et al. '06 is addressed. This family describes the dewetting process of nanoscopic thin liquid ?lms on hydrophobized polymer substrates due to the presence of several intermolecular forces and takes account of different ranges of slip-lengths at the polymer substrate interface. Reduced ODE models derived from underlying lubrication equations allow for an efficient analytical and numerical investigation of the latest stage of the dewetting process: coarsening dynamics of the remaining droplets. We present a new geometric approach which can be used for an alternative derivation and justi?cation of above reduced ODE models and is based on a center-manifold reduction recently applied by Mielke and Zelik '08 to a certain class of semilinear parabolic equations. One of the main problems for a rigorous justi?cation of this approach is investigation of the spectrum of a lubrication equation linearized at the stationary solution, which describes physically a single droplet. The corresponding eigenvalue problem turns out to be a singularly perturbed one with respect to a small parameter ε tending to zero. For this problem we show existence of an ε-dependent spectral gap between a unique exponentially small eigenvalue and the rest of the spectrum.
Dec 15, 2009 SFB 647
Jan 12, 2010 SFB 647
Jan 26, 2010 Prof.Dr. Sergei Shapiro
(Freie Universität Berlin, FR Geophysik)
Fluid-induced seismicity and pore-pressure diffusion in rocks
Dr. Marc Timme
(MPI Dynamics&Selforganization, Göttingen)
Spatio-Temporal Patterns in the Brain: Neural Networks with Nonlinear Dendrites
Feb 2, 2010 Prof. Dr. Flavio Abdenur
(PUC, Rio de Janeiro)
Geometric mechanisms for robust transitivity
14:15, Free Unversity Berlin, Room 140, Arnimallee 7 (rear building), 14195 Berlin
The existence of robustly transitive but non-Anosov diffeomorphisms has been known ever since Shub's construction in the early seventies. In the past 15 years or so, new examples have been given, many of which rely on the blender studied by Diaz and Bonatti. In addition, for the first time general necessary properties of such systems have been obtained: "robust transitivity implies such-and-such a condition". In this talk I will briefly survey the literature and then discuss some ongoing work that goes in the opposite direction: we ask which conditions are sufficient for - and hopefully in some sense characterize - the phenomenon of robust transitivity. This involves ongoing joint work with S. Crovisier.
SFB 647
Feb 9, 2010 Dr. Przemyslaw Perlikowski
(Technical University of Lodz)
Dynamics in a ring of delayed coupled Stuart-Landau oscillators
Results presented in the talk consider the ring of unidirectionally delayed coupled Stuart-Landau oscillators. Firstly, we show investigations of steady state stability, we present its properties with respect to delay in coupling and number of oscillators in the ring. To cope with small and large delay in the coupling we applied analytical techniques, while for intermediate delay we use Lambert function. Then, we present the influence of time delay and size of the ring on appearance of periodic solutions. We find coexisting stable periodic solutions (here, rotating waves), which correspond to Eckhaus instability. Presented results are ongoing joint work with S. Yanchuk.

Time and Place

Talks usually take place on Tuesday at 3:15 p.m.
at the Weierstraß Institute
Ehrhard-Schmid-Hörsaal, Mohrenstr. 39, 10117 Berlin.

Tea and coffee will be served at 2:45 p.m. on the ground floor.
Guests are always welcome !


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