Nonlinear Dynamics at the Free University Berlin

Summer 2012

# Oberseminar Nonlinear Dynamics

## Program

 Apr 17, 201216:15 Bernhard Brehm(Free University Berlin) Formal Kasner-map shadowing and qualitative results on the convergence of Bianchi 9 trajectories to heteroclinic chains The Bianchi 9 cosmologies have a well known compact heteroclinic attractor. Heteroclinic chains on this attractor are described by orbits of the mixing, non-injective Kasner map (related to continued fraction expansions). It has been long conjectured, that the dynamics of the Kasner-map (i.e. the structure of the heteroclinic network) essentially describes the long-term dynamics of trajectories which limit to the attractor. Unfortunately, it is not clear at all what we should precisely mean by this heuristic. The choice of definitions is here certainly a matter of taste as well as a tradeoff between attainability, readability and strength (for particular applications) of results. Aside from global (e.g. statistical or global topological) properties, one important notion is what we should mean by convergence of two trajectories towards each other or towards a heteroclinic chain. I will introduce the notion of convergence in order'', apply it to a classic example, and show some theorems pertaining to this choice of specification of the Kasner-map heuristic. Furthermore, I will argue that this notion of convergence sits at a good spectral gap'' for Bianchi 9 cosmologies: The definition is weak enough to allow for relatively easy results, while beeing strong enough that these results are still meaningful. Unfortunately the notion of convergence is too weak to answer many important questions; however, any strengthening of the notion of convergence seems to make similar results much harder to prove or outright wrong. (well, there \emph{is} a reason for some conjectures on Bianchi cosmologies still being unresolved after fifty years) The notion of convergence in order will be used to show that every (non-Taub) trajectory in Bianchi 9 converges in order'' to a (possibly non-unique) formal heteroclinic chain of the Kasner-map. Furthermore, we will show (if time permits) that for every infinite heteroclinic chain, there exists a codimension one set of initial conditions converging to it (by degree-type arguments). We will close by explaining why this result is not at all as spectacular as it might sound at first glance. Apr 24, 2012 Nadezhda M. Ratiner(Voronezh State University) Some applications of the topological degree for Fredholm proper maps with positive index to differential equations The talk will be devoted to some examples demonstrating applications of the topological degree for Fredholm proper maps with positive index. Nonlinear boundary value problem for system of ordinary differential equations, bifurcation problem for elliptic boundary value problem and oblique boundary value problem will be considered. The talk is partially based on a joint paper with V. G. Zviagin. May 08, 2012 SFB 647 May 15, 2012 Serhiy Yanchuk(Humboldt University Berlin) Patterns in non-homogeneous rings of delay coupled systems May 22, 2012 Fengqi Yi(Harbin Engineering University, China) Hopf bifurcations and steady state bifurcations of some semi-linear reaction diffusion equations Some semi-linear reaction diffusion equations subject to Neumann boundary conditions on bounded domains are considered. Hopf and steady state bifurcation analysis are carried out in details. In particular, for the concrete problems, we show the existence of multiple spatially nonhomogeneous periodic orbits while the system parameters are all spatial homogenous. Our results and global bifurcation theory also suggest the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions of certain reaction-diffusion systems. These results provide theoretical evidences to the complex spatio-temporal dynamics found by numerical simulation. May 29, 2012 SFB 647 Jun 05, 2012 SFB 910 Jun 12, 2012 Jens Starke(Technical University of Denmark) Traveling waves and oscillations in particle models The macroscopic behaviour of microscopically defined particle models is investigated by equation-free techniques where no explicitly given equations are available for the macroscopic quantities of interest. We investigate situations with an intermediate number of particles where the number of particles is too large for microscopic investigations of all particles and too small for analytical investigations using many-particle limits and density approximations. By developing and combining very robust numerical algorithms, it was possible to perform an equation-free numerical bifurcation analysis of macroscopic quantities describing the structure and pattern formation in particle models. The approach will be demonstrated with examples from traffic and pedestrian flow. The presented traffic flow on a single lane highway shows besides uniform flow solutions also traveling waves of high density regions. Bifurcations and co-existence of these two solution types are investigated. The approach is validated by a comparison with analytical findings. The pedestrian flow shows the emergence of an oscillatory pattern for two crowds passing a narrow door in opposite directions. The oscillatory solutions appear due to a Hopf bifurcation. This is detected numerically by an equation-free continuation of a stationary state of the system. Furthermore, an equation-free two-parameter continuation of the Hopf point has been performed to investigate the oscillatory behaviour in detail using the door width and relative velocity of the pedestrians in the two crowds as parameters. This is in parts joint work with Rainer Berkemer, Olivier Corradi, Yuri Gaididei, Poul Hjorth and Mads Peter Soerensen. Jun 19, 2012 SFB 647 Jun 26, 2012 Atsushi Mochizuki(RIKEN Advanced Science Institute, Japan) Pattern formation in biology I introduce two mathematical studies for pattern formations in biology. The first topic is on network and spot patterns in plants. Various differentiation processes in plant development are regulated by a plant hormone, auxin. Leaf vascular networks and phyllotaxis patterns in meristem are generated from network-like and spot-like distributions of auxin, respectively. Inhomogeneous distribution of auxin is generated by anisotropic distribution of auxin efflux carrier protein called PINFORMED (PIN) in cells. We develop mathematical models for the dynamics of auxin and PIN distribution in plant organs, and discuss possible mechanisms for switching between two different patterns, network-like and spot-like patterns. The second topic is on a simplest system of cell differentiation and pattern formation by a species of bacteria. A multicellular filamentous cyanobacterium, Anabaena, produces specialized nitrogen-fixing cells named heterocysts that appear about every 10 cells. We develop a 1-dimensional cellular automaton model for the dynamics of cyanobacteria, which includes stochastic cell division and differentiation. We determined distribution of heterocyst interval analytically. From the comparison with experimental data we conclude that age dependency of division and differentiation is essential for the observed patterns. This is joint work with Yoshinori Hayakawa and Jun-ichi Ishihara. Jun 19, 2012 SFB 647

## Time and Place

Talks usually take place on Tuesday at 3:15 p.m.
at the Free University Berlin
Arnimallee 7 (rear building), room 140.

Tea/coffee at 2:45 p.m. on the same floor.
Guests are always welcome !

## Archive

Last change: Oct. 11, 2012
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