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.
November 30, 2017 Nikita Begun (Saint Petersburg State University) Dynamics of Systems with Discontinuous Hysteresis Operator
We consider a two-dimensional dynamical system which couples linear equation with discontinuous hysteresis operator. Reduction to a Poincare map represents a piecewise linear discontinuous one-dimensional map $T$. We show that the dynamics of this map is conjugated with interval exchange maps on the unit circle. The seminar starts at 15:30.
December 07, 2017 Yuya Tokuta (Free University Berlin) On an extended model of Euglena Bioconvection
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 consider an extended model to incorporate the angular configuration of each cell of Euglena. The seminar starts at 15:30.
December 14, 2017 Ignacio Gonzalez (Free University Berlin) Genericity, transversality and applications
In my study of generic results in coupled cell networks of ODE's i have to deal with many of the main theorems that i will introduce today and they are applied in an almost identical way as in the most general generic results for ODE's. That's why introducing generic results without the framework of the networks it will maker easier to the audience to understand future results in the area of genericity in couple cell networks of ODE's.
Nicola Vassena (Free University Berlin) On how to invert matrices, and play with expansions and contractions of determinants
Finally, I have completely understood a proof by Brehm/Fiedler, which it is at the core of chemical networks sensitivity theory and that I was supposed to have understood more or less two years ago. In this talk I want to explain what I understood for the following reasons:
  • checking if I really have understood (nobody knows....)
  • looking for ideas and advices on how to modify/improve the proof but above all:
  • It's Christmas, and everybody should be more generous! That is why I want to share with you, my dear friends, my newly acquired (hopefully!?) deep knowledge.
January 11, 2018 Mark Curran (Free University Berlin) Approximating a reaction-diffusion equation with hysteresis using slow-fast systems with a diffusing slow variable
We consider a reaction-diffusion equation with an ensemble of hysteresis operators defined at every spatial point. We associated to this PDE in one variable, a system consisting of one PDE and an ensemble of ODEs defined at every spatial point. More specifically, we replace each of the individual hysteresis operators in the original PDE with a Van der Pol oscillator. I will outline a scheme showing that as the parameter in the Van der Pol oscillators goes to zero, the PDE-ODE system approximates the original PDE in the sup norm. In particular, I will show that the speed of convergence in the sup norm is proportional to this parameter raised to the power of two thirds, just as in the classical case of a single Van der Pol oscillator. The bulk of the talk will focus on how to deal with spatial points where the hysteresis operator is close to switching, or equivalently, where the corresponding Van der Pol oscillator is close to a saddle node bifurcation.
January 18, 2018 Ignacio Gonzalez (Free University Berlin) Introduction to transversality
In differential equations is very important to assert that a given property is generic with respect to some parameters, which may be the vector fields, the domains, etc. The transversality theorems are the usual way to do this. In this talk we will introduce the idea of transversality, we will prove the main theorems and we will see a few examples of how to use these tools. The talk will be sketched as follows: 1.Introduction, 2.Definitions and examples, 3.Stability of transversality, 4.Thom's transversality theorem, 5. Applications.

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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