Nonlinear Dynamics at the Free University Berlin

Summer 2021

Oberseminar Nonlinear Dynamics



Tuesday, April 20th Dr. Alan David Rendall (Johannes Gutenberg-Universität Mainz) Bogdanov-Takens bifurcations and the regulation of enzymatic activity by autophosphorylation
An important mechanism of information storage in molecular biology is the binding of phosphate groups to proteins. In this talk we consider the case of autophosphorylation, where the protein is an enzyme and the substrate to which it catalyses the binding of a phosphate group is that enzyme itself. It turns out that this often leads to more complicated dynamics than those seen in the case where enzyme and substrate are distinct. We focus on the example of the enzyme Lck (lymphocyte-associated tyrosine kinase) which is of central importance in the function of immune cells. We study a model for the activation of Lck due to Kaimachnikov and Kholodenko and give a rigorous proof that it admits periodic solutions. We do so by showing that it exhibits a generic Bogdanov-Takens bifurcation. This is an example where this approach gives a simpler proof of the existence of periodic solutions than ones using more elementary techniques. Joint work with Lisa Kreusser.
Tuesday, April 27th Eddie Nijholt (University of Illinois) Exotic symmetry in networks
Network dynamical systems appear all throughout science and engineering. Despite this prevalence, it remains unclear precisely how network structure impacts the dynamics. One very successful approach in answering this question is by identifying symmetry. Of course, there are many networks that do not have any form of symmetry, yet which still show remarkable dynamical behavior. Instead a wide array of other network features (such as node-dependency, synchrony spaces, and so forth) are known to impact the dynamics. We will see that most of these features can still be captured as symmetry, provided one widens the definition. That is, instead of considering classical group symmetry, one has to allow for more ``exotic structures'', such as semigroups, categories and quivers. In many cases the network topology itself can even be seen as such a symmetry. An important consequence is that network structure can therefore be preserved in most reduction techniques, which in turn makes it possible to analyse bifurcations in such systems. In order to best explain these notions, l do not assume any familiarity with group symmetry -or their exotic counterparts- on the part of the audience.
Tuesday, May 4th Dr. Maximilian Engel (FU Berlin) Lyapunov exponents in random dynamical systems and how to find and use them
This talk aims to give an overview on various notions of Lyapunov exponents (LEs) in random dynamical systems, that is, systems whose evolution in time is governed by laws exhibiting randomness: from finite-time LEs to classical asymptotic LEs and corresponding spectra up to LEs for processes conditioned on staying in bounded domains. We demonstrate how these notions, especially of a first, dominant LE, become relevant in the context of stochastic bifurcations, in finite and infinite dimensions.
Tuesday, May 11 Alicia Dickenstein (University of Buenos Aires) Algebra and geometry in the study of enzymatic networks?
I will try to show in my lecture that the question in the title has a positive answer, summarizing recent mathematical results about signaling networks in cells obtained with algebro-geometric tools.

Time and Place

Because of the current Corona virus situation, the events take place online.

Guests are always welcome !


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