Nonlinear Dynamics at the Free University Berlin

Winter 2016/17

Oberseminar Nonlinear Dynamics



Oct 25, 2016 SFB 647
Nov 1st, 2016 Bernhard Brehm
(Freie Universität Berlin)
Particle Horizons in the Mixmaster Universe
The Mixmaster Universe has been proposed by Misner (1969) as a model for a chaotic big bang cosmological singularity. This cosmological model describes Bianchi IX spatially homogeneous, anisotropic vacuum space-times. In 1970, Belinskii, Khalatnikov, and Lifshitz (BKL) conjectured that particle horizons form towards the big bang. In other words, backwards light-cones remain spatially bounded, and spatially separate regions causally decouple.

We prove this BKL conjecture, for almost every solution.

More specifically, the answer to this question depends on the convergence speed towards the Mixmaster attractor. Ringström (2001) showed that this convergence occurs at all. We introduce a novel expanding measure in order to prove that the convergence is fast enough to guarantee the formation of particle horizons for Lebesgue almost every solution.

The talk is addressed at a nonspecialist audience.
Nov 8, 2016 SFB 647
Nov 10, 2016 Prof. Sebastian van Strien
(Imperial College London, UK)
Dynamics on heterogenous networks
Networks in which some nodes are highly connected and others have low connectivity are ubiquitous (they are used to model the brain, the internet, cities etc). In this talk I will consider coupled dynamics on networks of this type. It turns out that weakly connected and highly connected nodes in this network often develop rather different kinds of dynamics, and that one can predict the behaviour for a time window which grows exponentially with the size of the network. These results are proved using ergodic theory and invariant manifolds. This work is joint with Matteo Tanzi and Tiago Pereira.

Please note that the seminar takes place at 14:15 in room 130, Arnimallee 3, 14195 Berlin
Nov 15, 2016 PD Dr. Martin Väth
(Freie Universität Berlin)
Abstract Volterra Equations
A definition of abstract Volterra operators/equations will be proposed which contains renewal and delay type equations. The relation between these problems will be illustrated and as a sample application a result about the essential spectral radius for linear abstract Volterra operators will be presented. Similar techniques apply also for the nonlinear case.
Nov 21, 2016 Dr. Jens Griepentrog
(Humdoldt Universität Berlin)
Solution regularity of parabolic variational inequalities
You are kindly invited to attend the special event on the occasion of the 65th birthday of Lutz Recke on Monday, November 21st, at 15:15 within the Research Seminar "Applied Analysis" of the Institute of Mathematics of the Humboldt University.

Please note that the seminar takes place on MONDAY at 15:15 in room 3.116 at Humboldt-Kabinett, Rudower Chaussee 25, 12489 Berlin-Adlershof

You are kindly invited to the seminar as well as to cake, coffee and wine afterwards. Guests are always welcome !
Nov 29, 2016 Prof. Dr. Tobias Jäger
(Friedrich-Schiller-Universität Jena)
Mode-locking phenomena in low-dimensional dynamics
Dec 6, 2016 Cancelled
Dec 20, 2016 Dr. Jan Sieber
(University of Exeter, UK)
Convergence of equation-free methods in the case of finite time scale separation
A common approach to studying high-dimensional systems with emergent low-dimensional behavior is based on lift-evolve-restrict maps (called equation-free methods originally proposed by IG Kevrekidis): first, a lifting operator maps a set of low dimensional coordinates into the high-dimensional space, then the high-dimensional (microscopic) evolution is applied for some time, and finally a restriction operator maps down into a low-dimensional space again. We prove convergence of equation-free methods for sufficiently large healing time (which is a method parameter). More precisely, if the high-dimensional system has an attracting invariant manifold with smaller expansion and attraction rates in the tangential direction than in the transversal direction (normal hyperbolicity), and restriction and lifting satisfy some generic transversality conditions, then the lift-evolve-restrict procedure generates an approximate map that converges to the flow on the invariant manifold for healing time going to infinity. In contrast to all previous results, our result does not require the time scale separation to be large. We demonstrate for an example from Barkley et al (SIAM J. Appl. Dyn. Sys. 5(3) 2006) that the ability to achieve convergence even for finite time scale separation is especially important for applications involving stochastic differential equations, where the evolution occurs at the level of distributions, governed by the Fokker-Planck equation. In these applications the spectral gap is typically small. In the example, the ratio between the decay rates of fast and slow variables is 2.
Jan 10, 2017 Dr. Christian Bick
(Somerville College, Oxford, UK/University of Exeter, UK)
From Weak Chimeras to Switching Dynamics of Localized Frequency Synchronization Patterns
We review some recent results on weak chimeras, a mathematically rigorous description for chimera states-solutions with coexisting synchronization and incoherence-in networks of identical oscillators. Moreover, we give some preliminary results on how to construct networks of identical phase oscillators with heteroclinic connections between weak chimeras of saddle type. These networks exhibit dynamic switching of localized synchronization which could for example encode information in neural networks.
Jan 24, 2017 Dr. Sergey Tikhomirov
(Saint-Petersburg State University, Russia)
Depinning bifurcation in quasiperiodic media
Jan 31, 2017 Prof. Matthias Wolfrum
(WIAS Berlin)
Interfaces in a chain of coupled bistable elements
Feb 7, 2017 Shalva Amiranashvili
(WIAS Berlin)
Scattering of waves at solitons
Feb 14, 2017 Alessandro Torcini
(Université de Cergy-Pontoise, Laboratoire de Physique Théorique et Modélisation)
Death and rebirth of neural activity in sparse inhibitory networks
In this presentation, we clarify the mechanisms underlying a general phenomenon present in pulse-coupled heterogeneous inhibitory networks: inhibition can induce not only suppression of the neural activity, as expected, but it can also promote neural reactivation. In particular, for globally coupled systems, the number of firing neurons monotonically reduces upon increasing the strength of inhibition (neurons' death). However, the random pruning of the connections is able to reverse the action of inhibition, i.e. in a sparse network a sufficiently strong synaptic strength can surprisingly promote, rather than depress, the activity of the neurons (neurons' rebirth). Thus the number of firing neurons reveals a minimum at some intermediate synaptic strength. We show that this minimum signals a transition from a regime dominated by the neurons with higher firing activity to a phase where all neurons are effectively sub-threshold and their irregular firing is driven by current fluctuations. We explain the origin of the transition by deriving an analytic mean field formulation of the problem able to provide the fraction of active neurons as well as the first two moments of their firing statistics. The introduction of a synaptic time scale does not modify the main aspects of the reported phenomenon. However, for sufficiently slow synapses the transition becomes dramatic, the system passes from a perfectly regular evolution to an irregular bursting dynamics. In this latter regime the model provides predictions consistent with experimental findings for a specific class of neurons, namely the medium spiny neurons in the striatum.

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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