Nonlinear Dynamics at the Free University Berlin

Summer 2024

Seminar Nonlinear Dynamics (Internal seminar)

Prof. Dr. Bernold Fiedler, PD Dr. Isabelle Schneider


Program



Thursday, April 18th Planning meeting
We are planning the seminar talks for this semester and will be giving the hybrid format a test run.
Thursday, May 2nd Danchen He (Freie Universität Berlin) Dynamics of an SEIR model with exogenous reinfections and quarantine control
Building upon the SEIR model proposed by Isaac Mwangi et al., our study analyzes the dynamic behavior of the SEIR model with exogenous reinfection influenced by quarantine control and transmission rates. In addition to the known backward bifurcation behavior, we prove that the model exhibits transcritical bifurcation when the basic reproduction number R0=1. Our results demonstrate the significant roles played by the disease transmission rate ß and the exogenous reinfection rate \tilde{p} (\tilde{p} is the exogenous reinfection rate influenced by quarantine) in altering the qualitative dynamics. We explore the global stability conditions of epidemic equilibrium under the joint influence of these two parameters. Furthermore, we investigate how our tuberculosis transmission model undergoes Hopf bifurcation under ß and \tilde{p}.
Thursday, May 23rd Willem van Zuijlen (WIAS Berlin) Anderson Models, from Schrödinger operators to singular SPDEs
The name Anderson model is used to refer either to the stochastic partial differential equation (SPDE) called the parabolic Anderson model or to the corresponding operator called the Anderson Hamiltonian. The operator is a random Schrödinger operator and in solid state physics this operator describes the evolution of a quantum state via the Schrödinger equation. On the other hand, the operator describes a random motion in a random environment by means of the parabolic Anderson model. Therefore the solution to both equations can be described by the spectral properties of the operator. I will discuss the Anderson Hamiltonian and the parabolic Anderson model with a white noise potential, which, due to its low regularity, brings us into the realm of singular SPDEs. One of the beauties of the parabolic Anderson model is the Feynman-Kac representation of the solution, by which one is able to derive the behaviour of its solution. I will motivate the model, give a feeling for the singularity and the construction of the Anderson Hamiltonian and describe how the relation to the parabolic Anderson is used to describe the asymptotic behaviour of its total mass.
Thursday, June 6th Mina Stöhr (WIAS Berlin / FU Berlin) Identical phase oscillators with global sinusoidal coupling and Möbius transformations
We discuss the dynamics of N identical phase oscillators with global sinusoidal coupling. These systems are known to display a simple form of collective behaviour. In this talk, we examine the underlying structure responsible for the low dimensional dynamics of these kinds of coupled oscillator systems and relate it to the group action of conformal mappings of the unit disk to itself - a subgroup of Möbius transformations.
Thursday, June 13th Johanna Rugen (FU Berlin) Kepler und die Weltharmonik
Note: This is a short student talk.

Thursday, June 20th

only online

Jia-Yuan Dai (National Chung Hsing University, Taiwan) See the invisible Ginzburg-Landau spiral waves: Galerkin control approach
Unstable solutions, while they exist theoretically, are often not observable in experiments or simulations. The challenge of " seeing the invisible " involves stabilizing unstable solutions in a noninvasive manner. In 1992, Pyragas introduced feedback control with time delays to stabilize ODE solutions. For PDE solutions, in 2022, Schneider developed symmetry-breaking control with spatio-temporal delays, with which we successfully stabilized certain classes of Ginzburg-Landau spiral waves. The primary limitation of symmetry-breaking control is its inability to distinguish solutions possessing the same symmetry, which hinders the stabilization of all spiral waves. To overcome this limitation, we propose using Galerkin control, which can differentiate more solutions than symmetry-breaking control. However, this approach requires additional information such as the bifurcation structure of spiral waves. The idea of proof for stabilization relies on the nodal property of spiral amplitudes. This is an ongoing research with Dr. Isabelle Schneider.

Thursday, June 27th

Nicola Vassena (Universität Leipzig) Mass action systems: Hopf without Hurwitz
Mass action systems identify parametric polynomial differential equations with a network structure. In general, it is a challenging question whether a mass action system admits a Hopf bifurcation. The standard tool is the Hurwitz method, which is however computationally expensive and thus limited to small networks. In contrast, I will address criteria for periodic orbits that do not involve any Hurwitz computation. The criteria involve three well-known ingredients, which somehow had not been properly mixed yet. The three ingredients are i) Linear algebra: D-stability and P-matrices, ii) a parametrization method for equilibria Jacobians of mass action systems, and iii) (global) Hopf bifurcation. Finally, I will present examples and give an interpretation of the underlying chemical mechanisms.

Thursday, July 4th

Phillipo Lappicy (Universidad Complutense de Madrid) Averaging near Taub points in Bianchi cosmologies
We consider vacuum spatially homogeneous Bianchi type VIII and IX models. In these ODE models, the Kasner circle of equilibria is a crucial object. It contains three special points with three null eigenvalues, called the Taub points. Hence, the Taub points possess a three-dimensional center manifold that coincides with the Bianchi type VI_0 and VII_0 models. We provide an approximating scheme in these invariant subspaces to describe the local dynamics nearby the Taub points via averaging methods.

Thursday, July 11th

Osman Bayram (Freie Universität Berlin) How the Logistic Map plays a role in the events of financial crashes
This is a short student talk.
Danchen He (Freie Universität Berlin) Dynamics of an SEIR model with exogenous reinfections and quarantine control
This is practice for a poster session.

Time and Place

Talks usually take place on Thursdays at 14:15, SR 140 in Arnimallee 7, 14195 Berlin

Guests are always welcome!


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