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.
Tuesday, June 1st Georgi Medvedev (Drexel University) Unfolding chimeras: Where Turing meets Penrose
The coexistence of coherence and incoherence is arguably the most interesting effect in the theory of synchronization and possibly in nonlinear science in general discovered in the past two decades. Despite intense research chimera states still present many challenging questions to the nonlinear science community. There is no consensus on how to define chimera states. Further, the theory is only available for chimera states lying in the Ott-Antonsen manifold, which is an elegant but a very special case. In this work, we suggest a new way for studying chimera states based on the combination of the linear stability analysis of mixing and a beautiful method of Penrose for Vlasov equation in plasma physics. This approach yields a new qualitative description of chimera states and provides very accurate quantitative estimates. Our results are universal in the sense that the structure and bifurcations of chimera states are explained in terms of the qualitative properties of the distribution of intrinsic frequencies and network topology, and, thus, are relevant for interacting particle systems of all scales from neuronal networks, to power grids, to astrophysics. This talk is based on the joint work with Hayato Chiba (Tohoku University) and Matthew Mizuhara (The College of New Jersey).
Tuesday, June 8th Chunming Zheng (MPI for Physics of Complex Systems, Dresden) Transition to Synchrony in the Three-Dimensional Noisy Kuramoto Model
We investigate the transition from incoherence to global collective motion in a three-dimensional swarming model of agents with helical trajectories, subject to noise and global coupling. Without noise this model was recently proposed as a generalization of the Kuramoto model and it was found, that alignment of the velocities occurs for arbitrary small attractive coupling. Adding noise to the system resolves this singular limit.
Tuesday, June 15th Philipp Lorenz-Spreen (MPI for Human Development, Berlin) Modeling radicalization dynamics and polarization in temporal networks
Echo chambers and opinion polarization have been recently quantified in several sociopolitical contexts, across different social media, raising concerns for the potential impact on the spread of misinformation and the openness of debates. Despite increasing efforts, the dynamics leading to the emergence of these phenomena remain unclear. Here, we propose a model that introduces the phenomenon of radicalization, as a reinforcing mechanism driving the evolution to extreme opinions from moderate initial conditions. Empirically inspired by the dynamics of social interaction, we consider agents characterized by heterogeneous activities and homophily. We analytically characterize the transition from a global consensus to an emerging radicalization that depends on parameters, which can be interpreted as the controversialness of a topic and the strength of social influence people exert on each other. Finally, we offer a definition of echo-chambers via our model and contrast the model's behavior against empirical data of polarized debates on Twitter, qualitatively reproducing the observed relation between users' engagement and opinions, as well as opinion segregation based on the interaction network. Our findings shed light on the dynamics that may lie at the core of the emergence of echo chambers and polarization in social media.
Tuesday, June 22nd Dr. Bhumika Thakur (Jacobs University, Bremen) Data driven identification of nonlinear dynamics using sparse regression with applications in plasma physics
Data driven techniques are increasingly finding applications in physical sciences and plasma physics is no exception. Many plasma processes are highly complex and nonlinear and often the exact form of the equations governing their dynamics is not known. If we can construct these equations from the experimental data, then we can further our understanding of these processes and use techniques such as model reduction to isolate dominant physical mechanisms. A large number of regression techniques are available for identification of system dynamics from data, with varying degrees of generality and complexity. Sparse identification of nonlinear dynamics (SINDy) algorithm is one such technique that can be used to find parsimonious models. I will talk about this algorithm and discuss some examples where it is being applied in plasma physics with a focus on our ongoing attempt at finding the model equations for anode glow oscillations observed in a glow discharge plasma device.
Tuesday, June 29th Sebastian Wieczorek (University College Cork) Rate-Induced Tipping Points
Many systems are subject to external disturbances or changing external conditions. For a system near a stable state (an attractor) we might expect that, as external conditions change with time, the stable state will change too. In many cases the system may adapt to changing external conditions and track the moving stable state. However, tracking may not always be possible owing to nonlinearities and feedbacks in the system. So far, the focus has been on critical levels of external conditions (dangerous autonomous bifurcation points) where the stable state turns unstable or disappears, causing the system to suddenly move to a different and often undesired state. We describe this phenomenon as bifurcation-induced tipping or B-tipping. However, critical levels are not the only critical factor for tipping. Some systems can be particularly sensitive to how fast the external conditions change and have critical rates: they suddenly and unexpectedly move to a different state if the external input changes too fast. This happens even though the moving stable state never loses stability in the classical autonomous sense! We describe this phenomenon as rate-induced tipping or R-tipping. Being a genuine non-autonomous bifurcation, R-tipping is not captured by the classical bifurcation theory and requires an alternative framework. In the first part of the talk, we demonstrate R-tipping in a simple ecosystem model where environmental changes are represented by time-varying parameters. We then introduce the concept of basin instability and show how to complement the classical bifurcation diagram with information on nonautonomous R-tipping that cannot be captured by the classical bifurcation analysis. In the second part of the talk, we develop a general mathematical framework for R-tipping with decaying inputs based on the concepts of thresholds, edge states and special compactification of the nonautonomous system. This allows us to transform the R-tipping problem into a connecting heteroclinic orbit problem in the compactified system, which greatly simplifies the analysis. We explain the key concept of threshold instability and give rigorous testable criteria for R-tipping to occur in arbitrary dimension. In the third part of the talk, we discuss the so-called ``compost-bomb instability", which is an example of R-tipping without an obvious threshold. We use geometric singular perturbation theory and desingularisation to reveal non-obvious R-tipping thresholds and edge states.
Tuesday, July 6th Prof. Dr. Carsten Conradi (Hochschule für Technik und Wirtschaft Berlin) Monomial parameterizations in the analysis of biochemical reaction networks
The dynamics of biochemical reaction networks can be described by ODEs with polynomial right hand side. In this presentation networks are considered where the steady state variety can be parameterized by monomials. I present two applications of these monomial parameterizations in the analysis of reaction networks: (i) deciding multistationarity and (ii) establishing Hopf bifurcations. Here multistationarity refers to the existence of at least two positive solutions to the polynomial steady state equations. And if a monomial parameterization exists, then this question is equivalent to the feasibility of at least one linear inequality system (out of many). The results presented here can be used to determine parameter values where multistationarity or Hopf bifurcations occur.
Tuesday, July 13th Prof. Dr. Messoud Efendiyev (Helmholtz Zentrum München) Mathematical modeling of biofilms and their long-time dynamics
In this talk we deal with mathematical modelling of biofilms, that show biofilm performance is non-uniform. Moreover, our model leads to a new class of degenerate PDEs. Effect of degeneracy to large time behavior of solutions will also be considered.

Time and Place

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

Guests are always welcome !


switch Last change: Jul. 7, 2021
This page strictly conforms to the XHTMLswitch1.0 standard and uses style sheets. Valid XHTML 1.0! Valid CSS!