Nonlinear Dynamics at the Free University Berlin

Winter 2022/2023

Forschungsprojekt Advanced Differential Equations

Prof. Dr. Bernold Fiedler, Isabelle Schneider


Thursday, October 27th Prof. Fiedler Concepts of time
Talk at the opening workshop of the Thematic Einstein Forum: ``Chaos and Contingency. The Role of Probability in Dynamical Models", Department of Physics, Studierendenzentrum BoB, Rooms 0.3.01 and 1.3.18, Arnimallee 14, 14195 Berlin. Online attendance is possible.
Thursday, November 3rd Phillipo Lappicy (Universidade Federal do Rio de Janeiro) Unbounded Sturm attractors for quasilinear parabolic equations
We construct the global attractors of quasilinear parabolic equations when solution can also grow-up (i.e., infinite time blow-up), in which case the attractor is unbounded with an induced nonlinear flow at infinity. In case of asymptotically semilinear diffusion, we prove that the induced flow at infinity is gradient and consists of unbounded equilibria with their unbounded heteroclinics orbits. Moreover, we explicitly prove the occurrence of heteroclinics between bounded and/or unbounded hyperbolic equilibria, noticing that the diffusion and reaction compete at infinity as regards the dimension of the unbounded dynamics.

at 17:15 Berlin time

Thursday, November 10th Prof. Bernold Fiedler (Free University Berlin) Sensitivity and Oscillations in Reaction Networks
We present some attempts to understand the dynamics of metabolic and gene regulatory networks, based on their graph structure, only. We describe steady state sensitivity to perturbed reaction rates based on joint work with Bernhard Brehm, and by Nicola Vassena. We also explore sufficient conditions for global Hopf bifurcation of time periodic oscillations. Specific examples include the citric acid cycle, and genetic circadian clocks in mammals. Talk at the NCTS Webinar on Nonlinear Evolutionary Dynamics

starting at 6:30 a.m. Berlin time via WebEx

Thursday, November 17th Isabelle Schneider (Freie Universität Berlin) Symmetry Groupoids in Dynamical Systems - Spatio-temporal Patterns and a Generalized Equivariant Bifurcation Theory
In this talk I will give an introduction and overview of my habilitation thesis. First, we redefine symmetry in dynamical systems and differential equations. Roughly speaking, symmetries are described by linear isomorphisms on flow-invariant linear subspaces which map solutions of dynamical systems to solutions. This allows us to prove generalized "equivaroid" versions of the equivariant branching lemma and the equivariant Lyapunov‒Schmidt reduction, as well as an extension of the equivaroid branching lemma which allows higher dimensional kernels. Second, we take a new perspective on spatio-temporal patterns of time-periodic solutions of differential equations. Specifically, we define spatio-temporal patterns as a groupoid acting directly on the space of periodic solutions, more precisely as a linear isomorphism which acts on the phase space and a component-wise time-shift. This construction gives us a new classification of spatio-temporal patterns beyond rotating and discrete waves. To prove the existence of such patterns, we generalize the equivariant Hopf bifurcation theorem, including a version which allows for higher dimensional kernels and thereby multi-frequency patterns. Lastly, we ask to construct dynamical systems with groupoid symmetries, where we focus on the rational design of networks with prescribed symmetries. Here we find that the relevant algebraic object is in fact the symmetry monoid paired to a given flow-invariant subspace.
December 8th Nicola Vassena (Freie Universität Berlin) Under construction
In this talk I want to share with the group a handful of messy and inconclusive thoughts concerning identification/exclusion of Hopf bifurcations in parametric systems.

Time and Place

Talks usually take place on Thursdays at 15:15
due to the current pandemic situation, our meetings will be held through Zoom.
If you wish to participate, please contact Alejandro for further details.

Guests are always welcome!


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