Nonlinear Dynamics at the Free University Berlin

Winter 2018/2019

Oberseminar Nonlinear Dynamics



Tuesday, October 16, 2018 Jan Sieber (University of Exeter) A nonsmooth saddle-node in a non-autonomous dynamical system"
Tuesday, November 6th Prof. em. Dr. Lutz Schimansky-Geier (Humboldt-University Berlin) Celestial mechanics of fruit flies or a theory for mushroomers
The topic of global search in complex environments have been often investigated. But a search can also be local in the sense that it is centered at a given home position. In the latter case, the searcher does not only look for a new target but is also required to regularly return to the home position. Such behavior is typical for many insects and achieves technical importance for self-navigating robotic systems. We propose a stochastic nonlinear model for local search which does not distinguish between the two aims. The dynamics bases on an unique pursuit and escape behavior of the heading from the position vector realizing thereby optimal exploration of space and the return to the home. We discuss the mechanics of the searcher and inspect the role of noise. Such randomness is present in the decision making rule of selecting the new heading direction. We consider Levy noises with different degree of discontinuity and report about steady spatial densities for the searchers. Also we report about an optimal noise intensity that a searcher finds a target at nearby places. For this noise value the required time for finding the target becomes minimal which appears to be the consequence of different relaxation processes in the spatial and the angular dynamics. Further extensions of the model are discussed during the lecture.
Tuesday, November 13th Hannes Stuke (FU Berlin) A dynamical systems approach to outlier robust machine learning
We consider a typical problem of machine learning - the reconstruction of probability distributions of observed data. We introduce the so-called gradient conjugate prior (GCP) update and study the induced dynamical system. We will explain the dynamics of the parameters and show how one can use insights from the dynamical behavior to recover the ground truth distribution in a way that is more robust against outliers. The developed approach also carries over to neural networks.
Tuesday, January 15th 2019 Prof. Willie Hsia (National Taiwan University) On the mathematical analysis of the synchronization theory with time-delayed effect
In this lecture, we will introduce a newly developed mathematical theory on the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase synchronization and frequency synchronization are in view. This is joint work with Bongsuk Kwon, Chang-Yeol Jung and Yoshihiro Ueda.
Tuesday, January 22nd 2019 Edgard Pimentel (Pontifical Catholic University of Rio de Janeiro) Geometric methods in regularity theory for nonlinear PDEs
In this talk we examine the regularity theory of the solutions to a few examples of nonlinear PDEs. Arguing through a genuinely geometrical method, we produce regularity results in Sobolev and Hölder spaces, including some borderline cases. Our techniques relate a problem of interest to a further one - for which a richer theory is available - by means of a geometric structure, e.g., a path. Ideally, information is transported along such a path, giving access to finer properties of the original equation. In the first part of the talk, we cover examples including elliptic and parabolic fully nonlinear problems, the Isaacs equation and a double divergence model. Then we proceed to the setting of state-dependent degenerate problems and report recent (optimal) results. We close the talk with a discussion on open problems and further directions of work.
Tuesday, January 29th 2019 Alejandro Kocsard (Universidade Federal Fluminense) Random walks, synchronization and existence of invariant measures
In this talk we start discussing some classical results due to Furstenberg [Fur63] about random product of matrices and how a certain form of synchronization naturally appears in this context. Then we shall consider some extensions of these results for random walks on compact smooth manifolds [Led84, Ant84, Cra90]. Finally we will see how these ideas can be used to study the existence of invariant (probability) measures for smooth actions on manifolds by some groups which are a priori non-amenable.
Tuesday, February 5th, 2019 Alejandro Kocsard (Universidade Federal Fluminense) Random walks, synchronization and existence of invariant measures II
In this second talk we will review some results we discussed in the first one, but from a "dynamical systems" point of view. Finally we shall discuss some applications of these results to the so called Burnside problem on manifolds.

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 !


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