Nonlinear Dynamics at the Free University Berlin

Summer 2017

Oberseminar Nonlinear Dynamics



Apr 25, 2017 Juliana Pimentel
(UFABC, Brazil)
Permutation characterization for slowly non-dissipative equations.
We study the longtime behavior of slowly non-dissipative scalar reaction-diffusion equations. By extending known results, we are able to obtain a complete decomposition of the non-compact global attractor and still manage to determine the heteroclinic connections based on the Sturm permutation method.

Please note that the seminar takes place at 3:15 p.m. at WIAS Berlin, Mohrenstr 39, in Erhard-Schmidt lecture room, 10117 Berlin.
Tea/coffee will be at 2:45 p.m. at WIAS as well.
May 2, 2017 Cancelled
May 9, 2017 Cancelled
May 16, 2017 Hannes Stuke
(Freie Universität Berlin)
Parabolic blow-up in complex time.
In my talk I will give an overview of my research on parabolic differential equations and blow-up. Blow-up denotes the phenomenon, when solutions diverge to infinity. By considering complex time we try to establish a connection between two opposed worlds: on the one hand bounded solutions, in particular equilibria and heteroclinic orbits and on the other hand blow-up in equations with entire nonlinearities.

Please note that the seminar takes place at 3:15 p.m. at WIAS Berlin, Mohrenstr 39, in Erhard-Schmidt lecture room, 10117 Berlin.
Tea/coffee will be at 2:45 p.m. at WIAS as well.
May 23, 2017 Pavel Gurevich
(Freie Universität Berlin)
A short introduction to machine learning: towards (un)certainty quantification.
Machine learning is a rapidly developing field that deals with searching for and generating patterns in data. It is nowadays a very broad field encompassing many tasks and methods. In my talk, I will give a short overview, rapidly narrowing my focus towards a particular topic, namely, neural networks and their ability not only to find patterns but also quantify how certain they are in doing so. If time permits, I will conclude with a discussion of our joint work with Hannes Stuke on certainty quantification.

Please note that the seminar takes place at 3:15 p.m. at WIAS Berlin, Mohrenstr 39, in Erhard-Schmidt lecture room, 10117 Berlin.
Tea/coffee will be at 2:45 p.m. at WIAS as well.
May 30, 2017 Phillipo Lappicy
(Freie Universität Berlin)
Einstein constraints: a dynamical approach.
The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a scalar quasilinear parabolic equation on the sphere, and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar solutions of Schwarzschild type, which describe the space of initial data of certain black holes, for example.

The first main result gives a detailed study of the axially symmetric solutions, since the domain is now one dimensional and nodal properties can be used to describe certain asymptotics of the rescaled self-similar solutions. Such asymptotics describe the possible metrics arising at an event horizon of a black hole, depending on the metric inside the horizon. Those are described by Sturm attractors. In particular, we compute an example for a prescribed scalar curvature.

The second main result states a symmetrization property of certain metrics in the event horizon, namely, how the symmetry of the spherical domain can influence the symmetry of solutions.

Please note that the seminar takes place at 3:15 p.m. in the lecture room 3.13, Hausvogteiplatz 11 A. Please wait at the doorkeeper's Mohrenstrasse 39 (usual location) to get access to the other building.
June 6, 2017 Cancelled
June 13, 2017 Cancelled
June 20, 2017 Cancelled
June 27, 2017 Igor Franović
(Institute of Physics Belgrade, University of Belgrade)
Bistability, rate oscillations and slow rate fluctuations in networks of noisy neurons with coupling delay
Spontaneous activity of cortical neurons is typically characterized as a doubly-stochastic process, underlying two distinct forms of variability. While the local spike-train variability is reflected on the fast timescale, the variability associated with much longer timescales involves macroscopic irregular fluctuations of the firing rate. The latter fluctuations apparently emerge by coherent switching of neurons between the “up” and “down” states of membrane potential, and are believed to play important functional roles. In order to gain qualitative insight into the mechanisms behind such switching phenomena, we consider a random network of rate-based neurons influenced by external and internal noise, as well as the coupling delay. The network behavior is analyzed by deriving the second-order stochastic mean-field model, which describes the network dynamics in terms of the mean-rate and the associated variance. The mean-field model is used to study the stability and bifurcations in the thermodynamic limit, as well as the fluctuations due to the finite-size effect. For the thermodynamic limit, it is established that (i) the network may exhibit coexistence between two stationary levels in a wide range of parameters, whereby the two types of noise affect the levels in a fundamentally different fashion, and (ii) coupling delay may give rise to oscillations of the mean-rate. The slow rate fluctuations are demonstrated to emerge via two distinct scenarios. In the delay-free case, the leading mechanism can be seen as noise-induced transitions between two metastable states, quite reminiscent to fluctuations of a particle in a double-well potential. In the second scenario, which involves the cooperative action of noise and delay, the fluctuations can be interpreted as stochastic mixing between two different oscillatory regimes.
July 4, 2017 Augusto Visintin
(Università di Trento)
Compactness and structural stability of nonlinear flows

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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