Nonlinear Dynamics at the Free University Berlin

Winter 2016/17

# Seminar Advanced Topics in Nonlinear Dynamics

Prof. Dr. Bernold Fiedler, PD Dr. Pavel Gurevich, Dr. Juliette Hell, Hannes Stuke

## Program

 Oct 20, 2016 Informal Meeting Planning Hannes Stuke(Freie Universität Berlin) Branching of analytic continuations of blow-up solutions The talk considers analytic continuations of blow-up solutions of the nonlinear quadratic heat equation. It was already shown by Masuda (87), that the blow-up point introduces branching of analytic continuations of solutions. I first try to describe possible branch types and in a second step I want to indicate how this can be used to construct actual solutions of a prescribed branch type. Furthermore I want to discuss how the branches introduced at the blow-up time are connected to asymptotics of solutions for large times. Oct 27, 2016 Isabelle Schneider(Freie Universität Berlin) Disputation: From Earth, Moon, and Sun to ∞ The three-body problem (Earth-Moon-Sun) has fascinated mankind for millennia. From ancient observations, first predictions and geometrical models, we take a journey to the modern formulation of the three-body problem as a system of ordinary differential equations: any three bodies moving under the influence of gravitation. In 2000, Chenciner and Montgomery discovered the first new exact periodic motions since Lagrange. In their "choreography", the three bodies chase each other along the same figure-eight curve. The main tools of their proof are the shape sphere and the principle of least action. The journey goes on … 15.15, Room SR 032, Arnimallee 6 Nov 03, 2016 Prof. Sergey Kryzhevich(St. Petersburg State University, Russia) On generalised shadowing An approach to find a weak form of shadowing is developed. We consider continuous maps of compact metric spaces. It is proved that every pseudotrajectory with sufficiently small errors contains at least one subsequence that can be shadowed by a subsequence of an exact trajectory with same indices. Later, we study homeomorphisms such that any pseudotrajectory can be shadowed by a finite number of exact orbits. We call this property multishadowing. Criteria for existence of epsilon-networks whose iterations are \epsilon networks are given. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed. Various applications of obtained results are given Prof. Alexandr I. Nazarov(St. Petersburg State University, Russia) On stability for time scale dynamical systems Abstract (PDF) Prof. Sergei Yu. Pilyugin(St. Petersburg State University, Russia) Clusters in dynamical systems A cluster in a dynamical system is a subsystem such that trajectories of the sybsystem have, in a sense, similar behavior. In this talk, we consider two types of clusters. In the first case, on selected time intervals, groups of variables are "freezed." It is shown that the resulting dynamics can be essentially more complicated than the dynamics of the "generating" system. In the second case, we "freeze" the dynamics of phase variables that do not belong to prescribed domains of the phase space. Aleksandr Enin(St. Petersburg State University, Russia) The multiplicity of positive solutions to a quasilinear Neumann problem with critical exponent Abstract (PDF) Aleksei V. Fadeev(St. Petersburg State University, Russia) Absence of inverse shadowing in actions of the Baumslag-Solitar groups We define the property of inverse shadowing for actions of finitely generated groups on metric spaces. It is shown that the action of the Baumslag-Solitar group BS(1,n) with n>1 on the real line does not have this property. Nikita Ustinov(St. Petersburg State University, Russia) Multiplicity of positive solutions to the boundary value problems with fractional Laplacians Abstract (PDF) Nov 10, 2016 Prof. Sebastian van Strien(Imperial College London, UK) Dynamics on heterogenous networks Networks in which some nodes are highly connected and others have low connectivity are ubiquitous (they are used to model the brain, the internet, cities etc). In this talk I will consider coupled dynamics on networks of this type. It turns out that weakly connected and highly connected nodes in this network often develop rather different kinds of dynamics, and that one can predict the behaviour for a time window which grows exponentially with the size of the network. These results are proved using ergodic theory and invariant manifolds. This work is joint with Matteo Tanzi and Tiago Pereira. Nov 17, 2016 Xiaobei Ma Analyzing Hopf Bifurcation and Bogdanov-Takens Bifurcation in Chemical Reactions We are inspired by an efficient algorithm to analyze the occurrence of Hopf Bifurcation in the chemical reaction systems, then try to improve the algorithm and expand it to check the condition of Bogdanov-Takens Bifurcation. We write the chemical reactions as polynomial dynamical systems using the mass-action modeling. The method transfer the concentration space to the reaction space based on stoichiometric network analysis to detect the bifurcation conditions on subsystem of flux cone. The condition is from the classical Routh-Hurwitz criterion. We will give MAPK cascade example to show the details. Michael Schäfer Denoising And Preserving Features In Grey Scale Images Nowadays, high definition pictures surround us nearly everywhere. The improvement of both quality and resolution has been accomplished by steady progress in image analysis. A major aspect concerns the removal of noise caused by the physical process of capturing images. On the other hand, significant features of the image should not be destroyed by this. Objects in nature usually belong to some hierarchical structure, corresponding to boundaries between different regions in the image. Both noise reduction and edge preservation are desired. Considering a grey scale image, diffusion equations are a valuable approach for noise reduction. However, the linear model does not restrict diffusion across distinct regions. The process needs to be controlled by information about edge locations. The gradient field seems to be a natural choice. The corresponding PDE formulation leads to adaptive denoising methods. References: "Scale-Space & Edge Detection Using Anisotropic Diffusion", Pietro Perona, Jitendra Malik, IEEE Transactions on Pattern And Machine Intelligence Vol. 12 No. 7, 1990 "Anisotropic Diffusion in Image Processing ", Joachim Weickert, B.G. Teubner Stuttgart, 1998 Nikita Begun The saw-like chaos in systems with hysteresis We consider a dynamical system with hysteresis. The system is motivated by modifications of general-equilibrium macroeconomic models that attempt to capture risks and memory dependence of realistic economic agents. Global dynamics and bifurcations of this system are studied depending on two parameters. We show that for a certain open set of parameter values, the system exhibits chaotic behavior. To understand the nature of this type of chaos, we introduce a map, which we call the “saw map”, and discuss its properties. Nov 24, 2016 Phillipo Lappicy (Free University Berlin) On the proof of liberalism We are interested in the dynamics of quasilinear parabolic differential equations with certain singularities at the boundary. In particular, the construction of the global attractor, which consists of equilibria and heteroclinic connections. In order to decide which equilibria are heteroclinically connected, three principles were proven in the semilinear case: cascading, blocking and liberalism. Those ideas will be recalled with focus on the proof of liberalism, including its adaptations for the quasilinear case with singularities at the boundary. Lastly, we will discuss yet another application of such global attractors. Dec 01, 2016 Adem Güngör (Free University Berlin) On chemical reaction network theory In my talk I would like to give a brief overview of my master thesis. I will introduce the area of chemical reaction network theory and give some tools, which are quite useful to analyse these dynamical systems. Alejandro López (Free University Berlin) Sturm attractors and spindles The introduction of the zero number in parabolic equations opened the gates to the study of the structure of the so-called Sturm attractors. The existence of a discrete Lyapunov function for delay differential equations with monotone feedback and the shape of some known examples make us question if both settings can be related. Yuya Tokuta (Free University Berlin) Bioconvection generated by Euglena Micro-organisms are known to form spatio-temporal patterns similar to those formed in the Rayleigh–Bénard model for thermal convection. Among such micro-organisms, Euglena gracilis form distinct patterns induced by positive/negative photo-taxis and sensitivity to the gradient of light intensity. A model for the convection patterns of Euglena gracilis was proposed by Suematsu et al. and we will discuss equilibria of the system. Mark Curran (Free University Berlin) Reaction-Diffusion Equations with Hysteresis in Higher Spatial Dimensions In this talk, we will treat a spatio-temporal problem where a diffusing substance is the input for a hysteresis operator defined at every spatial point. Such equations have been used to numerically model a number of biological processes exhibiting pattern formation. Though simulations agree closely with experiment, questions about the existence and uniqueness of solutions, as well as a rigorous explanation of pattern formation remain largely open. Well-possedness was only recently addressed on a one-dimensional domain. I will consider the problem in a higher-dimensional domain, and present a geometric condition on the initial data that ensure the problem in well-posed whenever such a condition is satisfied. Moreover, at any point in space, the hysteresis operator can be in one of two configurations, which naturally segregates the domain into two sub-domains. The concentration of the diffusing substance defines a switching mechanism, therefore our equation can be considered a type of free boundary problem. I will also describe the problem from this perspective, and relate it to a line of ongoing research. Jan 12, 2017 Paul Dieckwisch (Free University Berlin) Global attractors in chemical reaction networks Ismail Yenilmez (Free University Berlin) Inviscid Burger's Equation In my talk I will introduce the concept of weak solution of the inviscid burgers equation. Since our solution blows up in finite time if the wave from the left is faster than the wave from the right, therefore we are looking for weak solution. After being familiar with the Rankine-Hugoniot-condition and the Entropy inequality, we apply these concepts to one initial condition. Daniel Lebede (Free University Berlin) Fold-Hopf bifurcation in the MAP kinase cascade This talk will be about bifurcations in the mitogen-activated protein (MAP) kinase cascade, especially about the current progress in finding a Fold-Hopf bifurcation in the orignal system since it was already proven that there does not occur any Fold-Hopf bifurcation in the truncated system modelled by a Michaelis-Menten scheme and the Huang-Ferrell model. Jan 26, 2017 Jan Totz (Technical University Berlin) Experimental observation of spiral wave chimeras in coupled chemical oscillators I will present a versatile setup based on optically coupled catalytic micro-particles, that allows for the experimental study of synchronization patterns in very large networks of relaxation oscillators under well-controlled laboratory conditions. In particular I will show our experimental observation of the spiral wave chimera, predicted by Kuramoto in 2003. This pattern features a wave rotating around a spatially extended core that consists of desynchronized oscillators. We study its existence depending on coupling parameters and observe a transition to incoherence via core growth and splitting. Feb 02, 2017 David Molle (Free University Berlin) Lyapunov Functions, Global Attractors, and Sunflowers In my talk I will give an overview about the motivation and the goals of my master’s thesis about the sunflower equation and its global attractor. Ignacio Gonzalez (Free University Berlin) Observation and inverse problems in coupled cell networks In the talk will introduce one of the main theorems from the paper of Romain Joly quoted in the title from which my Master thesis will arise. Daniel Sarmiento Ferrera (Free University Berlin) Delayed Burger's equation I will talk about the topic of my master thesis. First, I show how the spectrum of the linearised operator around equilibria changes with respect to where the delay is placed in the equation. Later on, I will comment on some tools to prove existence of travelling wave solution to delayed reaction diffusion equations which may be adapted to my case. This is ongoing research. Alejandro Lopez (Free University Berlin) Transversality in delay equations Automatic transversality is one of the main tools for the study of Sturm global attractors. It is not known yet if spindles hold such a convenient property, throughout the talk we will study the behaviour of solutions close to hyperbolic equilibria and periodic orbits of delayed monotone feedback equations and observe the similarities with the parabolic problem and the limitations that may arise in this very different setting. Feb 09, 2017 Jia-Yuan Dai (Free University Berlin) Existence of rigidly-rotating and frozen spiral wave solutions of the lambda-omega system We prove that rigidly-rotating and frozen spiral wave solutions of the lambda-omega system exist. Our approach is based on a functional setting rather than the shooting method used in the most literature. The proof consists of two steps: 1. we apply the global bifurcation results of P. Rabinowitz to obtain frozen meridian wave solutions. 2. we apply the implicit function theorem to find genuine spiral wave solutions. It is worthy noting that our approach allows us to solve the existence problem for the lambda-omega system with Robin boundary conditions and complex diffusion rates.

## Time and Place

Talks usually take place on Thursday at 2:15 p.m.
at the Free University Berlin
Room 130, Arnimallee 3 (rear building), 14195 Berlin.

Guests are always welcome!

## Archive

Last change: Feb. 7, 2017
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