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NDNS+

Workshop

Nonlinear Dynamics of Natural Systems+

Eindhoven University of Technology
April 13 to April 16, 2010


Programme / Abstracts

Tuesday, April 13, 2010

10.30 Opening  
10.50-11.30 Robert Planqué Bird song dialects in homogeneous landscapes
11.30-12.10 Marianne Jonker Statistical Methods for Localizing Disease Genes
12.10-14.00 Lunch  
14.00-14.40 Vivi Rottschäfer Singular solutions of the generalised Korteweg-de Vries equation
14.40-15.20 Roeland Merks De novo and sprouting blood vessel growth: role of stochastic cell motility 
15.20-15.50 Break  
15.50-16.30 Ernst Wit Living in a Sparse World
16.30-17.10 Federica Pasquotto Symplectic invariants and existence of periodic orbits of Hamiltonian dynamical systems
17.10-17.50 Sorin Pop Non-classical travelling wave solutions to dynamic capillarity models

Wednesday, April 14, 2010

09.00-09.40 Jens Rademacher

Existence, stability and interaction of some nonlinear waves

09.40-10.20 Adrian Muntean

A two-scale RD system for gas-liquid reactions with nonlinear micro-macro transmission conditions: well-posedness and fast-reaction asymptotics

10.20-10.50 Break  
10.50-11.30 Huib de Swart Nonlinear tidal dynamics in channels with wide flat areas, an analytical model
11.30-12.10 Holger Waalkens

Classical and Quantum Reaction Dynamics in Multidimensional Systems

12.10-14.00 Lunch  
14.00-14.20 Shinji Nakaoka Daphnia revisited: an example of local stability and bifurcation analyses for physiologically structured population models
14.20-14.40 Robert Nolet Existence of traveling waves in the Diffusive VSC model
14.40-15.00 Sebastiaan Janssens The Circulator, a Hybrid Discrete-Continuous Time Model for Size-Structured Semelparity
15.00-15.30 Break  
15.30-15.50 Geert Geeven Computational statistics for the identification of transcriptional gene regulatory interactions
15.50-16.10 Kundan Kumar Modeling of LPCVD on the microstructured surface for 3D all-solid-state-battery applications
16.10-16.30 Rikkert Hindriks

Uncovering deterministic and stochastic dynamics from macroscopic neuronal recordings

17.00 Excursion & dinner  
  Willi Jäger After dinner lecture

Thursday April 15, 2010

09.00-09.40 Arjan van der Schaft Port-Hamiltonian dynamics on graphs
09.40-10.20 Daan Crommelin Stochastic subgrid scale modeling
10.20-10.50 Break  
10.50-11.30 Frank Redig Correlation inequalities in interacting particle systems
11.30-12.10 Jan Bouwe van den Berg Matched asymptotics for the harmonic map heat flow
12.10-14.00 Lunch  
14.00-14.20 Martin van der Schans Stability of blowup solutions in the Ginzburg-Landau equation
14.20-14.40 Michiel Renger A Microscopic Interpretation of the Entropy Increase Rate
14.40-15.00 Jaap Eldering Persistence of normally hyperbolic invariant manifolds in Banach spaces using the Perron method
15.00-15.30 Break  
15.30-15.50 Daniël Worm

Chemotaxis models in a space of measures

15.50-16.10

Michelangelo Vargas Rivera

Formal asymptotics for blowup in the Willmore flow
16.10-16.30 Alef Sterk The dynamics of a low-order model for the Atlantic Multidecadal Oscillation
16.30-16.50 Sjors van der Stelt Busse balloons: from the reversible Gray-Scott model to nonreversible Klausmeier models with nonlinear diffusion

Friday April 16, 2010

09.00-09.40 Renato Vitolo Robust Extremes in Chaotic Deterministic Systems
09.40-10.20 Sander Hille

Reverse engineering of the gradient sensing system in Dictyostelium requires stochastic modeling

10.20-10.50 Break  
10.50-11.30 Hans Heesterbeek Mathematics against infectious diseases
11.30-12.10 Stephan van Gils Dynamics of the basal ganglia
12.10-12.50 José Carrillo de la Plata Some kinetic models in swarming
12.50-14.00 Closing/Lunch  

ABSTRACTS

Jan Bouwe van den Berg

Matched asymptotics for the harmonic map heat flow

The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We will introduce the model and discuss some of its mathematical properties. In particular, we will focus on the possibility that singularities may develop. The rate at which singularities develop is investigated in settings with certain symmetries. We use the method of matched asymptotic expansions and identify different scenarios for singularity formation. More specifically, we distinguish between singularities that develop in finite time and those that need infinite time to form.

Presentation


José A. Carrillo de la Plata

Some kinetic models in swarming

I will present a kinetic theory for swarming systems of interacting, self-propelled discrete particles. Starting from the the particle model, one can construct solutions to a kinetic equation for the single particle probability distribution function using distances between measures. Moreover, I will introduce related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other non-trivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. I will also present and analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale, which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in Ha and Tadmor is obtained from a particle model. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of Cucker-Smale will be shown to hold for the solutions on the kinetic model. More precisely, the solutions concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.

Presentation


Daan Crommelin

Stochastic subgrid scale modeling

In atmosphere-ocean science, the representation (or parameterization) of subgrid scale processes in numerical models is a notoriously difficult problem. Good representations (or lack thereof) of processes such as convection, cloud physics and surface interactions directly affect the quality and accuracy of forecasts and climate studies. In recent years, researchers have turned to stochastic methods in order to improve these parameterizations. I will discuss a stochastic, data-driven approach to the problem, in which unresolved processes are represented by collections of (data-inferred) Markov processes that depend on resoved model variables. This approach leads to hybrid stochastic-deterministic models consisting of differential equations coupled to Markov chains.

Presentation


Jaap Eldering

Persistence of normally hyperbolic invariant manifolds in Banach spaces using the Perron method

The classical results on persistence of normally hyperbolic invariant manifolds by Fenichel and Hirsch, Pugh and Shub assume the invariant manifold to be compact. Since then, numerous improvements have been made, e.g. to semiflows in Banach spaces, but still assuming the invariant manifold to be compact.

We employ the Perron method to prove persistence of non-compact, infinite-dimensional normally hyperbolic invariant manifolds for continuous systems in Banach spaces. We obtain optimal smoothness results that include a H\"older continuity of the highest derivatives, exhausting the gap condition. That is, let the gap condition satisfy $\lambda_- = r\,\lambda_+$, with $\lambda_- < \lambda_+ < 0$ spectral bounds of the flow along the normal and tangential directions of the invariant manifold respectively. Then the invariant manifold is $C^{k,\alpha}$ for $k+\alpha < r$.

Presentation


Geert Geeven

Computational statistics for the identification of transcriptional gene regulatory interactions

Condition-specific and time-dependent transcriptional regulatory networks underlie the coordinated expression of genes involved in all biological processes. Insight into these networks is crucial for the understanding of biological systems under normal and pathological conditions. In this talk I will discuss how statistical models can be used to infer relationships between DNA binding proteins and target genes by analyzing experimental gene expression and DNA sequence data.

Presentation


Stephan van Gils

Dynamics of the basal ganglia

The basal ganglia consists of a number of strongly connected brain nuclei. In Parkinson’s disease, depletion of the neurotransmitter Dopamine affects the behavior of the Striatum, and subsequently all nuclei show pathological behavior. We have developed a model for a relay Thalamus cell to investigate the effects of the pathological basal ganglia output on the Thalamus. We will also report on very recent results for an animal model for Parkinson’s disease. In several brain nuclei local field potentials have been measured simultaneously. The results for normal rats and Parkinsonian rats are compared. The goal of this work is to optimize deep brain stimulation based on analysis of a network model for the basal ganglia.

Presentation


Hans Heesterbeek

Mathematics against infectious diseases

The epidemiology, dynamics and control of infectious diseases is an example of an important field of research where the use of mathematical reasoning is essential to gain understanding. It is, however, not only the biomedical field that benefits from this interaction; the problems that are studied frequently inspire new mathematical methods and results that are interesting in their own right. In this lecture I will give a number of examples where mathematics has increased our insight into infectious disease dynamics, and I will highlight a few areas where new mathematical input is needed. Malaria, influenza (Mexican and other), worm parasites, and plague are infections that will be used in the examples.

Presentation


Sander Hille

Reverse engineering of the gradient sensing system in Dictyostelium requires stochastic modeling

 The unicellular slime mould Dictyostelium discoideum is able to sense shallow gradients of the compound cyclic AMP in its environment and coordinate its movement machinery such that it quickly moves towards the source. Receptors on its cell membrane detect the compound. Together with an internal signaling network they convert the external signal in a proper, amplified, internal representation in terms of secondary messenger molecules. In this presentation we introduce the structure of this biochemical  mechanism, experimental results from biophysics on the movement of single receptors over the membrane and initial deterministic modeling, using a system of reaction diffusion equations.

We present joint work with Johan Dubbeldam (TUD) on simulation of this model in comparison to an appropriate stochastic model. This reveals that one needs to go beyond determistic modeling in order to gain proper understanding of the structure of this system and its functioning.

Presentation


Rikkert Hindriks

Uncovering deterministic and stochastic dynamics from macroscopic neuronal recordings

To elucidate the functional role of neuro-electrical processes in perception and cognition, a profound understanding of the dynamics that govern these processes is required.  In this talk we will concentrate on data-driven approaches to uncover dynamical principles underlying macroscopic neuronal recordings like magneto-encephalography (MEG) and local field potentials. Since electrophysiological processes are inevitably subject to stochastic fluctuations, their observed dynamics might be well described by stochastic differential equations. Although stochastic differential equations have been used to model time-series of several natural and biological systems, their application to neurophysiology is still very limited. In this talk we discuss a number of statistical issues that arise when fitting stochastic differential equations to time-series, such as the construction of consistent estimators for drift and diffusion functions, how to obtain confidence intervals, and how model verification can be performed. Moreover, we apply the methodology to MEG recordings of human subjects and to local field potentials recorded from mouse hippocampus in vitro. The results lead to a more profound insight into the dynamical principles underlying macroscopic neuronal processes.  

Presentation


Sebastiaan Janssens

The Circulator, a Hybrid Discrete-Continuous Time Model for Size-Structured Semelparity

In population biology a species is called semelparous if individuals reproduce exactly once during their lifetime and die immediately thereafter. Age-structured semelparous population models typically take the form of a non-linear matrix recurrence, as in the work of J.M. Cushing, N. Davydova and others.

In this talk we consider a situation in which size rather than age determines the moment of semelparous reproduction. Following up on a paper of W. Huyer from the mid-nineties, we consider a hybrid discrete-continuous time model which essentially takes the form of a time-T map pertaining to an ordinary differential equation composed with a map modeling the reproductive event. We focus on the existence, stability and bifurcation of single cohort solutions, i.e. Dirac measures. In our attempts to "lift" Huyer's finite-dimensional results to the infinite-dimensional setting of positive measures, we encounter several functional analytic and measure theoretic difficulties when trying to justify our numerical observations. Possible routes toward resolution will be discussed.

Presentation


Marianne Jonker

Statistical Methods for Localizing Disease Genes

In my talk I will explain two methods for localizing disease genes: case- control association analysis and linkage analysis. In a case-control association study one tries to find genes that cause a disease by comparing the genomes of affected and healthy individuals (cases and controls). In a linkage study family data is considered. Here, the idea is that if a disease runs in a family, a chromosomal region that runs exactly the same way in the family contains the causal gene. I will show how statistics can be helpful in the search for disease-genes. Foreknowledge on genetics is not necessary, since I start my talk with a short introduction into genetics.

Presentation


Kundan Kumar
Joint work with I.S. Pop, J.F.M. Oudenhoven, M.H.J.M. de Croon, P.H.L. Notten

Modeling of LPCVD on the microstructured surface for 3D all-solid-state-battery applications

We present a model for low pressure chemical vapor deposition (CVD) on a surface having microstructures in the form of trenches. Inside the trenches, the diffusion process is described by the Knudsen diffusion, as the free molecular path length of the molecules is comparable to the characteristic length scale of the trenches. Furthermore, in case of thin trenches one cannot neglect the influence of the deposited material on the geometry of trenches. As the deposition is depending on the unknown solution, the position of the interface separating the deposited layer from the gas is also unknown and becomes a part of the problem. Hence, the problem is a free boundary problem. This talk is addressing modeling details related to the gas flow and the deposition process, as well as the coupling conditions between the two components of the model. Further we present the computational technique for simulating the CVD, including the deposition inside the trenches. This technique is inspired by domain decomposition, and couples the bulk flow with processes inside trenches. Using steady state approximations, we obtain reasonable estimates for the physical and chemical parameters. Arbitrary Lagrangian-Eulerian (ALE) method is used to compute the solution for the free boundary problem on the microscale. The modeling is validated by comparison with experimental results.

Presentation


Roeland Merks

De novo and sprouting blood vessel growth: role of stochastic cell motility 

Controlling the growth of blood vessels is crucial for numerous biomedical problems ranging from tumor angiogenesis and diabetes to tissue engineering. One of the key players of blood vessel growth are endothelial cells, which form the internal “wall paper” of our blood vessels. When cultured in a gel mimicking their natural environment, endothelial cells assemble into networks that resemble the embryonic blood vessel network.  

To find out how they can do that, we developed a simulation model based on observed behaviors of endothelial cells. The model uses the cellular Potts model, which naturally represents cells' stochastic cell behaviors.  In the simplest models, endothelial cells secrete a molecular signal that attracts surrounding cells; hence they assemble into isolated clusters, not networks. We will discuss a series of additional assumptions by which cells do assemble into blood-vessel-like networks, and form blood vessel sprouts (angiogenesis).  Our models suggest that molecular biasing of stochastic membrane extensions and retractions may be crucial for network formation and blood vessel sprouting, as well as for other branching processes in development biology. 

Presentation


Adrian Muntean
Joint work with Maria Neuss-Radu (Heidelberg)

A two-scale RD system for gas-liquid reactions with nonlinear micro-macro transmission conditions: well-posedness and fast-reaction asymptotics

I consider a coupled two-scale (micro-macro) nonlinear reaction-diffusion system
modelling gas-liquid reactions. The novel feature of the model is the nonlinear transmission condition coupling the microscopic and macroscopic concentrations, given by a nonlinear Henry-type transfer function. The solution is approximated by using a Galerkin method adapted to the multiscale form of the system. This approach leads to the existence and uniqueness of the solution, and can  be used for numerical computations of  a larger class of nonlinear multiscale problems. Finally, I show that in the fast-reaction limit the two-scale system converges to a two-scale free-boundary problem (with the free boundary traveling the microstructure).

Presentation


Shinji Nakaoka
Joint work with Odo Diekmann, Mats Gyllenberg, Hans Metz and Andre de Roos

Daphnia revisited: an example of local stability and bifurcation analyses for physiologically structured population models

We present analytical and numerical techniques to investigate stability and bifurcation properties of a system of delay equations, a couple of renewal equation and delay differential equation. A size-structured resource consumer population model which represents a population growth of a Daphnia population is investigated.

Presentation


Robert Nolet

Existence of traveling waves in the Diffusive VSC model

The Diffusive Vesicle Supply Center (VSC) model attempts to explain the growth of fungal hyphae. Hyphae are hairlike structures, of fixed diameter and great length, on certain fungi. Some basic assumptions on the growth yield a free boundary problem for the evolution of the hyphal cell wall. In this talk we will use a Schauder fixed point approach to show the existence of traveling wave solutions exhibiting the observed behavior of the fungi.

Presentation


Federica Pasquotto

Symplectic invariants and existence of periodic orbits of Hamiltonian dynamical systems

A famous conjecture by A. Weinstein, dating back to 1979, states that periodic orbits of Hamiltonian dynamical systems must always exist on compact energy hypersurfaces of so called "contact type", a geometric property related to the symplectic structure carried by the phase space of the system under consideration. This conjecture has been established as a theorem in a number of interesting cases, at first using variational methods, more recently by constructing some very powerful symplectic invariants - based on Gromov's theory of J-holomorphic curves - which are able to detect the existence of periodic orbits. In this talk I will give an overview of the main results, try to sketch the basic ingredients for the construction of the invariants and in particular focus on joint work with R. Vandervorst concerning their possible extension to the case of non-compact energy hypersurfaces.

Presentation


Robert Planqué

Bird song dialects in homogeneous landscapes

Many bird species, especially song birds but also for instance some hummingbirds and parrots, have noted dialects. By this we mean that locally a particular song is sung by the majority of the birds, but that neighbouring patches may feature different song types.

Behavioral ecologists have been interested in how such dialects come about and how they are maintained for over 45 years. As a result, a great deal is known about different mechanisms at play, such as dispersal, assortative mating and learning of songs, and there are several competing hypotheses to explain the dialect patterns known in nature. There is, however, surprisingly little theoretical work testing these different hypotheses at present. We will start with the simplest kind of model in which one may speak of dialects, and which takes into account the most important biological mechanisms: one where there are but two patches, and two song types. It teaches us that a combination of little dispersal and strong assortative mating ensures dialects are maintained. However, contrary to first intuition, song learning has either a neutral or a negative impact on the maintenance of dialects.

Presentation


Jens Rademacher

Existence, stability and interaction of some nonlinear waves

I give an overview of my main research done with support of the NDSN+ cluster at the CWI. It concerns nonlinear travelling waves and related coherent structures in spatially one-dimensional reaction- diffusion-type equations, and in so-called Fermi-Pasta-Ulam atomic chain models.

Various stability and interaction issues are discussed, in particular absolute and convective instabilities of wave trains in the PDEs and boundaries of stability regions. The methods range from rigorous analysis and formal asymptotic matching to numerical computations.

Presentation 1

Presentation 2


Frank Redig

Correlation inequalities in interacting particle systems

The symmetric exclusion process (SEP) is a system of particles performing independent random walks where jumps that lead to more than one particle per site are forbidden. This "repulsive" (fermionic) interaction leads to the fact that in the SEP particles are more spread out than independent random walkers. This is rigorously expressed in the so-called Liggett's comparison inequality. As a consequence, product measures evolve in the SEP towards measures with negative correlations. Recently, a natural attractive analogue of the SEP, the so-called symmetric inclusion process (SIP) has been introduced. Here, jumps that tend to bring particles together are favoured rather than prohibited. We prove the reverse Liggett's inequality for this system and prove that local Gibbs measures evolve towards measures with positive correlations. As a consequence, positivity of correlations in some non-equilibrium steady states is obtained. By duality, this property also holds for a related system of interacting diffusion processes.

Presentation


Michiel Renger

A Microscopic Interpretation of the Entropy Increase Rate

It is well known that for small time steps, solutions of the diffusion equation optimize an energy functional containing entropy and the Wasserstein distance. We built upon this result by studying how the functional arises as the limit of microscopic particle systems. To this aim we consider a finite number of independent Brownian particles and calculate the jump probability in a fixed time. The theory of large deviations allows us to calculate a rate functional from the jump probability that for small time converges to the Wasserstein-Entropy energy functional.

Presentation


Vivi Rottschäfer

Singular solutions of the generalised Korteweg-de Vries equation

In this talk we study singular solutions of the generalised Korteweg-de Vries equation (KdV). The stability of solitary waves of the KdV has already been studied extensively. These solitons can become unstable and become infinite in finite time, in other words blow up. We analyse the structure of these blowup solutions. After introducing a dynamical rescaling the solutions are found as bounded solutions to an ODE. We study this ODE using asymptotic methods to construct the solutions.


Martin van der Schans

Stability of blowup solutions in the Ginzburg-Landau equation

In this talk we study the stability of radially symmetric blowup solutions of the Ginzburg-Landau equation (GL). The GL is an amplitude equation, which is derived as model in for example: nonlinear optics, models of turbulence, Rayleigh-B\'enard convection, superconductivity, superfluidity, Taylor-Couette flow and reaction diffusion systems Since these derivations lose their validity for large amplitude, it is of particular interest to study existence and stability of blowup solutions to the GL.

Blowup solutions are solutions of the GL of which the norm blows up in finite time. Existence and numerical stability have already been studied. Particularly surprising are numerical stability results of certain radially symmetric ring-like solutions , since in many equations, for example the nonlinear Schrödinger equation and the nonlinear heat equation, ring solutions were found to be unstable. We study linear stability of blowup solutions of the GL with analytic methods.

As a first step in the stability analysis we consider radially symmetric perturbations and focus on linear stability. To obtain linear stability results we use the asymptotic construction of the solution and Evans function techniques, which is a standard technique for analyzing eigenvalue problems. Due to the nature of the asymptotics Evans function techniques are not directly applicable.

Presentation


Arjan van der Schaft

Port-Hamiltonian dynamics on graphs

A topic of great current interest, motivated by diverse applications, is the subject of dynamics on networks. In this talk we discuss how one can define generalized Hamiltonian dynamics (possibly including resistive elements, algebraic constraints, and external ports) on graphs in an intrinsic way. Main tool in this endeavor is the definition of two Dirac structures determined by the graph. (A Dirac structure is a geometric object generalizing at the same time symplectic forms and Poisson brackets.)

The first Dirac structure, dating back to the classical work by Kirchhoff, is the appropriate Dirac structure for e.g. defining the dynamics of RLC electrical circuits in a port-Hamiltonian way. In this case the dynamics is associated to the edges of the graph. The second Dirac structure allows to associate dynamics to every vertex of the graph, and is the natural Dirac structure to formulate e.g. the dynamics of consensus algorithms for multi-agent systems, or the dynamics resulting coordination control strategies. If time permits we will also discuss the possibility to extend this framework to one-complexes, or hypergraphs, thereby also covering chemical reaction networks.

Next to the modeling and analysis of port-Hamiltonian dynamics on graphs we will discuss ways to synthesize or control the port-Hamiltonian dynamics on graphs, and to reduce the complexity of the dynamics in a structure

Presentation


Sjors van der Stelt

Busse balloons: from the reversible Gray-Scott model to nonreversible Klausmeier models with nonlinear diffusion

The Klausmeier model is a nonreversible reaction-diffusion-advection model for vegetation patterns in semi-arid systems on sloped terrains. In flat terrains, the model becomes reversible: the advection term is taken over by a diffusion effect. In an oversimplified setting, this diffusion is linear and the resulting equation is up to rescaling identical to the Gray-Scott system (for auto-catalytic reactions). In a more realistic setting, the diffusion term is nonlinear (since it is related to groundwater flow) and the resulting equation is can be seen as a `porous medium Gray-Scott equation'. In this talk, we will discuss existence and dynamics of stable spatially periodic patterns in this family of  porous medium Klausmeier/Gray-Scott models. The discussion will be based on existing results on spatial patterns in the Gray-Scott model and will be centered around the concept of the Busse balloon -- the (bounded) region in (parameter, wave number)-space for which stable spatially periodic patterns exist. The talk will be a mixture of an analytical approach -- we will classify the nature of the onset of pattern formation by Ginzburg-Landau techniques -- and computational methods based on continuation software.

Presentation


Sorin Pop
Joint work with C.J. van Duijn, Y. Fan (Eindhoven), L.A. Peletier (Leiden) and P.A. Zegeling (Utrecht)

Non-classical travelling wave solutions to dynamic capillarity models

We consider an extended Buckley-Leverett (BL) equation describing two-phase flow in porous media. This equation includes a third order mixed derivatives term modeling dynamic effects in the capillary pressure. For this extended equation we investigate the existence of traveling wave solutions. In analogy with the classical theory for hyperbolic conservation laws, we consider the limit case when capillary effects are vanishing. This leads to admissible shocks for the original BL equation, which violate the Oleinik entropy condition and are therefore called nonclassical. This allows constructing non-monotone weak solutions of the BL problem that consist of steady states separated by shocks, which are ruled out by standard two phase flow models, but agree with results obtained experimentally.

Presentation


Alef Sterk

The dynamics of a low-order model for the Atlantic Multidecadal Oscillation

Observations and model studies have provided ample evidence for the presence of multi-decadal variability in the North Atlantic sea-surface temperature. This variability is now known as the Atlantic Multidecadal Oscillation (AMO). We will show that it is possible to derive a low-dimensional dynamical system which captures the AMO in terms of a periodic orbit which is born through a Hopf bifurcation when a damping parameter is decreased. Further bifurcations of this periodic orbit are studied using techniques and concepts from dynamical systems. In the product of state and parameter space we have coexistence of periodic dynamics, quasi-periodic dynamics, and chaos.

Everything occurs in a physically `visible' way (i.e., occurs with positive measure in parameter space).

Presentation


Huib de Swart
Joint work with J.F.F. Zimmerman, Institute for Sear Research

Nonlinear tidal dynamics in channels with wide flat areas, an analytical model

Characteristics of tidal motion in channels are strongly influenced by nonlinear processes. In particular, the latter cause asymmetry of tidal curves, both of sea surface and velocity. This asymmetry has profound implications for net sediment transport and thereby for the morphodynamic stability of the channel. Sources of nonlinearities are advection of momentum, bottom stresses and depth-dependent friction, but also exchange processes between the channel and tidal flat areas along the sides of the channel. Previous studies have shown that, in order to understand observed tides in e.g. the Tamar Estuary, it is necessary to account for the fact that tidal flats act as a temporal storage of mass. In this contribution the focus is on yet another phenomenon, viz., tidal flats also act as a net sink term for along-channel momentum. This momentum loss term is a highly nonlinear function of sea surface elevation and velocity, and is significant in areas where the width of tidal flats, b, is of comparable or larger size than the width B of the main channel. The importance of momentum loss will be demonstrated by employing a perturbation method to a nonlinear shallow water model. This results in approximate, analytical solutions that describe the principal tide, overtides, as well as residual sea surface and currents. The process of momentum loss will be analysed, and its efficiency in generating nonlinear tides will be compared with other sources of nonlinearities. By comparing model output with field data it will be shown that momentum loss generates measurable asymmetries of the tide.


Michelangelo Vargas Rivera

Formal asymptotics for blowup in the Willmore flow

The Willmore flow is a model for the bending energy of bio-membranes. It is an open question whether the Willmore flow can produce a singularity in finite time on a smooth surface. We use matched asymptotics and moving mesh methods to investigate this problem.

Presentation


Renato Vitolo

Robust Extremes in Chaotic Deterministic Systems

Abstract: The prediction of disruptive or catastrophic events (in e.g. the climate system or financial markets) is of great scientific and societal interest. Extreme value theory (EVT) provides models and methods for statistical prediction in stationary stochastic processes. Significant progress has been recently obtained in extending EVT to chaotic deterministic systems. Although the behaviour of such systems is often very sensitive to variation of control parameters (bifurcations), in certain cases the statistics of extremes smoothly depends on external (control) parameters. Two examples and the outline of a theory are presented in this talk. Future research ideas will be discussed, building on current joint research with H. Broer, A. Sterk (Univ. Groningen), H Dijkstra (Univ. Utrecht) and C Simo' (Univ. Barcelona).


Holger Waalkens

Classical and Quantum Reaction Dynamics in Multidimensional Systems

A system displays reaction type dynamics if its phase space possesses bottleneck type structures. Such a system spends a long time in one phase space region  (the region of 'reactants'), and occasionally finds its way through a bottleneck to another phase space region (the region of 'products'), or vice versa. In Hamiltonian systems such bottlenecks are induced by equilibrium points of saddle-center-...-center type ('saddles' for short). The most widely used method to compute reaction rates is Transition State Theory which has its origin of conception in chemistry where it was invented by Wigner, Eyring and Polanyi in the 1930's. The main idea here is to compute the reaction rate from the flux through a dividing surface placed in the bottleneck (or in chemical terms 'transition state') region. In order not to overestimate the rate the dividing surface needs to have the so-called 'no-recrossing' property which means that it is crossed exactly once by reactive trajectories and not crossed at all by nonreactive trajectories. The construction of such a dividing surface has posed a major problem in Transition State Theory since its invention in the 1930's. In the first part of my talk I will discuss in detail the phase space structures which govern the dynamics `across' saddles, and how they can be computed from a normal form.

This implies the construction of a dividing surface without recrossing which, as we will see, is 'spanned' by a normally hyperbolic invariant manifold. In the second part of the talk I will discuss the implications of the classical phase space structures for the quantum mechanics of reactions. We will see that a quantization of the classical normal form leads to an efficient algorithm to compute quantum reaction rates and the associated quantum resonances.

The talk summarizes joint work with Roman Schubert and Stephen Wiggins from Bristol University.

Presentation


Ernst Wit

Living in a Sparse World

As the result of the ever decreasing cost of measuring and storing data, the amount of real information – defined as non-spurious relationships between variables – has become ever more sparse. Traditional low dimensional inferential schemes, whereby one aimed to learn about the world by means of multiple observations for each parameter, are being replaced by high-dimensional ones, where the number of parameters far outstrip the number of observations.

The advent of high-dimensional datasets has presented a challenge to traditional statistical inference. The n>p paradigm turned out to be too restrictive and statisticians seemed to be for a while in high seas.

However, they found their (wet) feet again, when they realized the connections between high-dimensional inference on the one hand and model choice and penalized methods on the other.

Presentation


Daniël Worm
Joint work with Sander Hille

Chemotaxis models in a space of measures

We start by discussing a kinetic chemotaxis model and briefly mention some results we achieved on global existence of positive mild solutions in (intersections of) L^p-spaces.

It would be of biological and mathematical interest to be able to extend the model to a space of measures. However, the total variation norm on this space is often too strong, so we introduce a different Banach space into which the measures are densely embedded and mention some of its properties. We also indicate how the chemotaxis model can be formulated in this space.

Presentation


Last updated 11-mei-2010,
By LC

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