Nonlinear Dynamics at the Free University Berlin

Winter 2019/2020

Oberseminar Nonlinear Dynamics



Tuesday, October 29th Pavel Gurevich and Hannes Stuke (Freie Universität Berlin) Reinforcement learning
Tuesday, November 5th Phillipo Lappicy (Universidade de Sao Paulo, Freie Universität Berlin) Horava-Lifshitz Gravity: Bifurcations and Chaos
Lorentzian causal structure, general covariance, and scale invariance are first principles that play a key role in the nature of generic spacelike singularities in general relativity. To bring a new perspective on the contributions of these first principles regarding the chaotic aspects of generic singularities, we consider the initial singularity in spatially homogeneous Bianchi type VIII and IX models in Horava-Lifshitz gravity, which replaces relativistic first principles with anisotropic scalings of Lifshitz type. To describe the nature of the initial singularity in these models, we make use of mathematical tools that include symbolic dynamics and chaos. For the present class of models, it is shown that general relativity is a critical case that corresponds to a bifurcation where chaos becomes generic. For different models nearby the general relativistic critical case, Cantor sets and iterate function systems are important for describing the chaotic aspects of generic singularities. This is joint work with Juliette Hell and Claes Uggla.
Tuesday, November 12th Nicolas Perkowski From hopping particles to stochastic PDEs
I will try to give an overview of some of my research interests, focusing on certain a priori ill posed stochastic PDEs and their derivation. An important task in stochastics is to find and construct "universal" models that describe a given phenomenon. For example, any random variable that is given by the superposition of many small independent influences is approximately Gaussian, independently of the concrete nature of the small influences, and therefore we call the Gaussian distribution universal. When trying to derive universal models for phenomena that evolve in space and time, formal calculations often suggest that we should consider nonlinear stochastic PDEs driven by space-time white noise. This is a problem because due to the irregularity of the noise the solution might be too irregular to make sense of the nonlinearities in the equation. But in recent years we found new ways of overcoming these problems, making sense of the equations, and proving their universality in some cases.
Tuesday, November 19th Ralf Toenjes (University of Potsdam) The Constructive Role of Noise in The Dynamics on Network Hubs for Network Synchronization
We describe and analyze a coherence resonance phenomenon for synchronization in bipartite networks of well connected hubs and followers when the hubs are subjected to noise. Using the Ott-Antonsen ansatz for globally coupled phase oscillators the dynamics of the mean fields is described by a low-dimensional system of Langevin equations. Averaging over the fast stochastic dynamics of the hubs yields ordinary differential equations which predict the coherence resonance reasonably well.
Tuesday, November 26th Jia-Yuan Dai (National Center for Theoretical Sciences) Hyperbolicity of Ginzburg-Landau vortex solutions
We prove that each equilibrium of the Ginzburg-Landau equation restricted on the invariant subspace of vortex solutions is hyperbolic, that is, its associated linearization possesses nonzero eigenvalues. This result completely describes the global attractor of vortex solutions, and also yields the Ginzburg-Landau spiral waves of nodal type. This result is a joint work with Dr. Lappicy.
Tuesday, January 14th Frits Veerman (University of Heidelberg) Toll roads and freeways: Defects in bilayer interfaces in the multi-component functionalised Cahn-Hilliard equation
We study a multi-component extension of the functionalised Cahn-Hilliard equation, which provides a framework for the formation of patterns in fluid systems with multiple amphiphilic molecules. The assumption of a length scale dichotomy between two amphiphilic molecules allows the application of geometric techniques for the analysis of patterns in singularly perturbed reaction-diffusion systems. For a generic two-component system, we show that solutions to the four-dimensional connection problem provide the leading order approximation for solutions to the full eight-dimensional barrier problem, which can be obtained through a perturbative expansion in the layer width. Moreover, we show that a saddle-node bifurcation of bilayer solutions in the four-dimensional connection problem acts as a source of so-called defect solutions, i.e. solutions to the barrier problem that are not also solutions to the connection problem. The analysis combines geometric singular perturbation theory with centre manifold theory in an infinite-dimensional context.
Tuesday, January 28th Martin Brokate (TU München) Sensitivity results for rate independent evolutions
Rate independent evolutions are evolutions whose solution operators commute with time transformations. They are inherently nonlinear and nonsmooth. We present some introduction, outlining various mathematical approaches as well as relations to other fields. We then address sensitivity, in particular the question whether the solution operators possess weak differentiability properties.
Tuesday, February 11th Peter Szmolyan (Technische Universität Wien) Cancelled due to illness Advances in GSPT -- teaching new tricks to an old dog
Due to the efforts of many people geometric singular perturbation theory (GSPT) has developed into a very successful branch of applied dynamical systems. GSPT has proven to be very useful in the analysis of an impressive collection of diverse problems from natural sciences, engineering and life sciences. Fenichel theory for normally hyperbolic critical manifolds combined with the blow-up method at non-hyperbolic points is often able to provide remarkably detailed insight into complicated dynamical phenomena, often even in a constructive way. Much of this work has been carried out in the framework of slow-fast systems in standard form, i.e. for systems with an a priori splitting into slow and fast variables. More recently GSPT turned out to be useful for systems for which the slow-fast structures and the resulting applicability of GSPT are somewhat hidden. Problems of this type include singularly perturbed systems in non-standard form, problems depending singularly on more than one parameter and smooth systems limiting on non-smooth systems as a parameter tends to zero. Often several distinct scalings must be used to cover the dynamics of interest and matching of these different scaling regimes is carried out by the blow-up method. In this talk I will survey these developments and highlight some ongoing activities.

Time and Place

Talks usually take place on Tuesday at 3:15 p.m. at Freie Universität Berlin, Arnimallee 3, Room 130, 14195 Berlin.

Tea/coffee will be served at 2:45 136.
Guests are always welcome !


switch Last change: Feb. 11, 2020
This page strictly conforms to the XHTMLswitch1.0 standard and uses style sheets. Valid XHTML 1.0! Valid CSS!