Nonlinear Dynamics at the Free University Berlin

Winter 2017/2018

Seminar Advanced Topics in Nonlinear Dynamics

Prof. Dr. Bernold Fiedler, Hannes Stuke


Program

October 19, 2017 Stephen Lynch (Free University, Berlin) Ancient solutions to extrinsic curvature flows
We will introduce the mean curvature flow of hypersurfaces and see how the study of its finite-time singularities naturally leads one to study solutions defined for all negative times, known as ancient solutions. Such solutions tend to enjoy rigidity results, since diffusion is allowed an infinite amount of time to homogenise their geometry. We will study the proof of one such result, due to Huisken and Sinestrari, which says that any compact ancient solution satisfying a certain uniform convexity condition must be a shrinking sphere. We will then discuss generalisations of this result to flows of hypersurfaces by speeds other than the mean curvature, and to the mean curvature flow of submanifolds of higher codimension.
October 26, 2017 Fustel Soh (Free University, Berlin) Spatio-temporal feedback control for partial differential equations on a two-dimensional domain.
As a critical example to the spatio-temporal delayed feeback control approach for partial differential equations, it is our goal to control of the solutions’ Stability for the heat equation, u_t = u_xx + u_yy + cu under the periodic boundary conditons on a two-dimensional domain. We classify the equilibria into unstable and stable and construct control terms to stabilize the unstable ones. We define necessary conditions to be satisfied by the control to ensure stability.Having inserted these control terms we do not change the equilbria in themselves but we get all equilibria to become stable.
Carlos Rocha (Instituto Superior Técnico of Lisbon) Design of Sturm attractors by shooting meanders
We use the permutation and shooting meander characterization of Sturm global attractors to address the realization of specific examples of Sturma attractors. In particular we consider the realization of snoopy planar attractors, and the 3-ball snoopy and bun attractor. We also address the impossibility of realization of the snoopy burger, a CW-complex with two 3-balls, as a Sturm global attractor.
November 2, 2017 Yuya Tokuta (Free University, Berlin) Time-averaging of reaction-diffusion equations with the application to Euglena Bioconvection under rapidly oscillatory illumination.
Microorganisms are known to form spatiotemporal patterns similar to those formed in the Rayleigh-Bénard model for thermal convection. Among such, Euglena gracilis form distinct patterns induced by phototaxes and sensitivity to the gradient of the light intensity. A model for the convection patterns under stationary illumination was proposed by Suematsu et al. and we will modify the model to discuss patterns under oscillatory illumination. We will cover time-averaging of the system in the case of rapid oscillation.
Carlos Rocha (Instituto Superior Técnico of Lisbon) Design of Sturm attractors by shooting meanders.
Continuation of the talk of the 24th October.
November 9, 2017 Jia-Yuan Dai (Free University, Berlin) Existence of local solutions of Gowdy spacetimes.
In this talk we consider a class of Gowdy spacetimes that reduces the Einstein’s field equation into a system of two semilinear wave equations, by assuming a universe without matter in which the gravitational wave fronts repeat in space and are mutually parallel. To prove the existence of local solutions of the system, we adopt a functional setting to seek periodic-in-space solutions. We show that the spectrum of the linearization operator around the trivial solution are nonresonant eigenvalues. Then we discuss a small-divisor problem related to the application of the Lyapunov- Schmidt reduction.
This ongoing research is a joint work with Hannes Stuke.
November 16, 2017 Nicola Vassena (Free University, Berlin) Sensitivity of chemical reaction networks: present and future.
In my talk I will give an overview about my current research topic and ideas for further developments. I will explain how our research relates to established existing literature and, in particular, focus on the issues regarding bifurcation analysis.
Phillipo Lappicy (University of Sao Paolo) An invitation to constructing attractors
This talk will discuss possible generalisations of the attractor construction which was done for quasilinear parabolic equations. We will focus on three particular cases: nonlinear diffusion of the p-laplacian type, nonautonomous equations and fully nonlinear parabolic problems. In particular, we will show that the latter has a Lyapunov function, by adapting a method of Matano.
November 23, 2017 Alejandro Lopez (Free University, Berlin) Periodic orbits in systems with monotone delayed feedback
In the talk existence and connectivity properties of periodic solutions of scalar equations with monotone delayed feedback will be discussed. I will cover my current topic and directions of research as well as what has been achieved to date.
Matteo Levi (Universita' di Bologna) Equilibrium measures on trees
On a metric space (X,d) one can define a set function called capacity, which has motivations coming from Physics and plays a deep role in potential theory and geometric measure theory. It is well known that to any compact subset E of X, it can be associated a probability measure on X called equilibrium measure for E.These measures at present are not well understood. We will present a characterization of equilibrium measures when X is a locally finite tree of infinite depth.
Stefano Vita (Universita' di Torino) Competition-diffusion elliptic systems
In this talk I will do an introduction to competition-diffusion elliptic systems. We can imagine this kind of sistems as a model for the interaction between populations which spread in space. If the populations are aggressive, we can imagine such interaction as a repulsion. In particular, as the parameter of aggressivity grows, the populations segregate their support, and a free boundary appears between their positivity sets. I am interested in the study of properties of the limiting profiles, and of the structure and regularity of the free boundary.

Time and Place

Talks usually take place on Thursday at 2:15 p.m.
at the Free University Berlin
Room 140, Arnimallee 7 (rear building), 14195 Berlin.

Guests are always welcome!


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