Nonlinear Dynamics at the Free University Berlin

Summer 2021

Seminar Advanced Delay Equations and Advanced Differential Equations

Prof. Dr. Bernold Fiedler, Dr. Isabelle Schneider


Thursday, April 15th Kick-off meeting -
After a short semester break, let us reconvene online! Since communication was severely hindered by the pandemic, let us start off with a short review of the past months: What did you do (research-wise, conferences, projects, etc)? Give us a short update! Let us also look forward to the upcoming summer semester: What are your plans and goals?
Thursday, April 22nd Nicola Vassena (Freie Universität Berlin) Column-swaps between matrices interconnected by a permutation structure
Motivated by bifurcation analysis on metabolic networks, we present an elementary linear algebra problem. For a MxM matrix, the principal minors of order (M-1) are the ones obtained by removing the i-th row and the i-th column, for i=1, … M. Given two MxM matrices A and B interconnected by a certain permutation structure, we characterize the determinants of all (M-1)x(M-1) matrices obtained by column-swaps between principal minors of order (M-1) of A and the corresponding principal minors of order (M-1) of B. With the intention of keeping the talk self-contained as long as possible, the motivation for such a curious problem is postponed at the end, if time consents.
Thursday, April 29th Dennis Chemnitz (Freie Universität Berlin) Periodic Solutions of Delay Equations on the circle
In this talk I will continue the analysis of periodic solutions with winding number 1 for delay equations on the circle. Specifically I will introduce a new viewpoint involving circle homeomorphisms and will talk about restrictions for possible secondary branches.
Thursday, May 6th Isabelle Schneider (Freie Universität Berlin) Symmetry in dynamical systems
I will present an extended notion of symmetry and show you some examples where the generalised symmetry provides new information on the solutions of a dynamical system.
Thursday, May 20th Hauke Sprink (Freie Universität Berlin) Stable manifolds for periodic cycles in Horava Lifshitz (HL) cosmology
In this talk we give a perspective on results by Liebscher et al who construct stable manifolds for a certain class of heteroclinic chains in the well-known Bianchi model in general relativity. The aim of our work is to obtain similar results in the parameter dependent BIanchi models within HL cosmology.
Thursday, May 27th Danchen He (Freie Universität Berlin) Properties of an epidemic model with control interventions
Phillipo Lappicy (Universidade de Sao Paulo (ICMC), Brazil) Subcritical chaos and averaging in Horava cosmologies
We will discuss recent progress in the bachelor theses of Victor Hugo and Giuliano Pantarotto within the context of Horava cosmologies. VH aims at proving chaos for the multivalued Kasner map for the subcritical case, when the parameter v is less than 1/2; whereas GP looks for a local description of the dynamics nearby the Taub points by means of a center manifold reduction and an averaging approximation through a truncated Hamiltonian system.
Thursday, June 3rd Alejandro López Nieto (Freie Universität Berlin) The Andronov-Pontryagin criterion for delay equations with negative feedback
In analogy to the 2-dimensional case, we discuss a structural stability criterion for the global dynamics of a delay equation with negative feedback.
Thursday, June 10th Tilman Glorius (Freie Universität Berlin) Period-Delay Mismatch in a Pyragas Controlled System
Si Mohamed Sah (Department of Mechanical and Electrical Engineering, The University of Southern of Denmark) Identification of damping and stiffness from output only data
In this work I plan to identify damping and stiffness from output only data using theoretical techniques. The results of this ongoing project could be helpful in the diagnosis of vibrating structures while they are under continuous operation.
Thursday, June 17th Babette de Wolff (Freie Universität Berlin) Delayed feedback stabilization with and without symmetry
Pyragas control is a time delayed feedback control scheme that is designed to stabilize periodic orbits. In situations where the dynamical system possesses symmetries, we can use equivariant control to select and stabilize a specific spatio-temporal pattern. In this talk, we discuss both limitations to Pyragas control and successful stabilization with equivariant control. We start by proving an invariance principle for Pyragas control. This invariance principle leads to a clear understanding of the well-known odd number limitation, and also implies a new `any number limitation'. We then discuss how these limitations to Pyragas control can be overcome using equivariant control, and in particular discuss a successful stabilization result for a class of discrete waves.
Thursday, June 24th Jia-Yuan Dai (National Center for Theoretical Sciences, Taiwan) Dynamics of two predators competing for a renewable prey (Part II)
We consider an ODE system that describes the dynamics of two predators competing for the same prey. Our aim is to extend the boundary limit cycle (i.e., a periodic solution when one of predators is absent) into the positive cone, as global as we can. For this aim, in this talk we discuss how the limit cycle becomes unstable and we also estimate its period. This is ongoing research with Dr. P. Lappicy, Dr. N. Vassena, and Dr. H. Stuke.
Thursday, July 1st, at 16:30 Berlin time Bernold Fiedler (Freie Universität Berlin) On meanders
Meanders describe Jordan curves which cross a horizontal axis transversely, at N points. They also relate to pairs of parenthesis stackings (Dyck words), folded strips of stamps, to traces in Temperley-Lieb algebras, certain Cartesian billiards, the shooting approach to nonlinear second order boundary value problems in ordinary differential equations (ODEs), and the geometry of global attractors of scalar partial differential equations of parabolic type (PDEs). But, even their asymptotic combinatorics remains open. We illustrate how the PDE global attractors of ODE meanders turn out to be regular Thom-Smale cell complexes, with an additional sign structure. Many examples will accompany our approach. We happily exploit contributions by Pablo Castaneda, Anna Karnauhova, Phillipo Lappicy, Alejandro Lopez Nieto, Carlos Rocha, Matthias Wolfrum, Peter Zograf, and many others. Theoretical flatworm chopping and grafting by Angela Stevens et al., however, presents a surprise.
Thursday, July 8th Isabelle Schneider (Freie Universität Berlin) Symmetry in dynamical systems - Part 2
I will continue my talk on a generalized notion of symmetry in dynamical systems. We will focus on the construction of prescribed patterns and on bifurcation.
Thursday, July 15th Isabelle Schneider (Freie Universität Berlin) Symmetry in dynamical systems - Part 3
I will continue my talk on a generalized notion of symmetry in dynamical systems. We will focus on the construction of prescribed patterns, periodic orbits, and the connection to equivariant Pyragas control.

Time and Place

Talks usually take place on Thursdays at 15:15
due to the current lockdown 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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