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Dynamic Analysis Seminar (DAS)

The Dynamic Analysis seminar is organized jointly with the analysis group at the UvA.

The seminar (usually) takes place every other Wednesday, 16:00-17:00.

Talks are intended for an audience with a background in analysis and dynamical systems, including PhD students.

For more information, please contact the organizers: Rob van der Vorst (VU), Bob Rink (VU), Han Peters (UvA) and Ale Jan Homburg (UvA).

A database of earlier years' seminars at the VU can be found here.

Upcoming talks in 2012

Currently, no more talks are scheduled for this semester.

Previous talks in 2011/2012

Wed. 25 April @ UvA, 16:00-17:00, Room A1.04: Bert Peletier (Leiden)
Title: The dynamics of "Target-Mediated Drug Disposition"
Abstract: Drugs are designed to interact with specific targets in order to produce their desired pharmacological effect. This involves a dynamic interplay between drug and target, each of which is supplied and eliminated, and the drug-target complex which is also eliminated or absorbed. In this talk we discuss the often nontrivial dynamics of this process, ways to recognise it experimentally, and attempts at developing simplified models.
Wed. 11 April @ VU, 16:00-17:00, Room M-632: Henk Broer (Groningen)
Title: Resonance and Fractal Geometry
Abstract: A number of resonant phenomena is reviewed such as Huygens's synchronizing clocks, the tidal resonances of Moon and certain planets as well a swing. Resonance is an interaction of various oscillations with rationally related frequencies which leads to a compatible periodic behaviour. It is conceived of in terms of parameter dependent dynamics. The resonant zones in parameter space then can consist of tongues that are arranged in a fractal pattern in which a Cantor set plays a role. In and near the Cantor set also other types of dynamics may occur, like quasi-periodic or chaotic. In the talk we discuss several examples.
Wed. 28 March @ UvA, 16:00-17:00, Room B0.209: Martijn Zaal (VU)
Title: Time discretization of the osmotic cell swelling problem
Abstract: A simple model for cell swelling by osmosis can be formulated as a free boundary problem involving diffusion and mean curvature. This problem can in turn be studied using a time discretization originating from the study of gradient flows in Euclidean space. This point of view relates the physics of the model to the mathematics used to construct solutions. Moreover, it illustrates how ideas from the theory of gradient flows can be used outside of Euclidean or even metric space.
Wed. 14 March @ VU, 16:00-17:00, Room M-632: Evgeny Verbitskiy (Leiden/Groningen)
Title: Periodic and homoclinic points in algebraic dynamics
Abstract: This will mainly be a gentle introduction to actions generated by a finite number of commuting automorphisms of compact abelian groups, and their fascinating connections with algebra, number theory, analysis, and algebraic geometry. I'll focus on the problem of the growth rate of periodic points for such actions, including recent joint work with Doug Lind and Klaus Schmidt. The main technical tool relies on construction of suitable homoclinic points of algebraic dynamical systems.

Wed. 29 Feb. @ UvA, 16:00-17:00, Room TBA: Yonatan Gutman (Warschau)
Title: The structure of cubespaces attached to minimal distal dynamical systems
Abstract: Cubespaces were recently introduced by Camarena and B. Szegedy. These are compact spaces $X$ together with closed collections of "cubes" $C^{n}(X)\subset {2^{n}}$, $n=1,2,\ldots$ verifying some natural axioms. We investigate cubespaces induced by minimal dynamical topological systems $(G,X)$ where $G$ is Abelian. Szegedy-Camarena's Decomposition Theorem furnishes us with a natural family of canonical factors $(G,X_{k})$, $k=1,2,\ldots$. These factors turn out to be multiple principlal bundles.We show that under the assumption that all fibers are Lie groups $(G,X_{k})$ is a nilsystem, i.e. arising from a quotient of a nilpotent Lie group.This enable us to give simplified proofs to some of the results obtained by Host-Kra-Maass in order to characterize nilsequences internally.

Wed 7 Dec @ UvA, 16:00-17:00, Room G0.05: Bob Rink (VU)
Title: A destruction theorem for generalized Frenkel-Kontorova crystal models
Abstract: The equilibrium states of the classical Frenkel-Kontorova crystal are also the orbits of the Chirikov standard twist map. In a generalized Frenkel-Kontorova crystal there is no such correspondence, because the atoms in such a crystal interact beyond their nearest neighbors. In this talk, I will present the following converse KAM theorem for generalized Frenkel-Kontorova crystals: if the crystal model admits a continuous family of ground states with an average particle spacing that is "easy to approximate by rational numbers", then this family can be destroyed by an arbitrarily small smooth perturbation of the crystal model. This means that a "typical" crystal will display "forbidden regions" for its atoms. This result of Blaz Mramor and myself generalizes a theorem of Mather for the destruction of Liouville invariant circles of twist maps. Our proof is quite different though and may allow for generalizations to lattice Aubry-Mather theory and elliptic PDEs

Wed 9 Nov @ UvA, 16:00-17:00, Room G0.05: Tom Kempton (Utrecht University)
Title: Bernoulli convolutions and Beta expansions
Abstract : In this talk we'll discuss two objects, one from number theory and one
from measure theory, and link them using some dynamics and ergodic theory.
The first object is the set of beta-expansions of a real number. Given
beta>1 and a real number x, a beta expansion of x is a sequence (a_n)
for which we can write x=\sum a_i\beta^{-i}.
If we put beta=10 and take a_i in the digit set {0,1,...,9} then beta
expansions are the familiar decimal expansions. Almost every x has a
unique decimal expansion, making the set of decimal expansions of a
given real number rather dull. However, for non-integer beta the set
of beta expansions of a given real number is typically much richer. We
are interested in the structure of this set.
The second object that we'll discuss is the Bernoulli convolution.
Bernoulli convolutions are perhaps the most simple examples of fractal
measures, and yet the fundamental question of whether the Bernoulli
convolution associated to some parameter beta is absolutely continuous
remains unsolved.
In this talk we'll link some questions about the structure of the set
of beta expansions of a typical real number with the question of
absolute continuity of the Bernoulli convolution.
Wed 26 Oct @ VU, 16:00-17:00, Room F-640: Pablo Barrientos
Title: Heteroclinic cycles arising in generic unfoldings of nilpotent singularities
Abstract : In this seminar we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we show that any nilpotent singularity of codimension four in R4 unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in R3 unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes (Bykov cycles). Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.
Wed 12 Oct @ UvA, 16:00-17:00, Room G0.05: Erlend Fornaess Wold (Oslo)
Title: Symplectic Completion of Jets
Abstract : We will show how one can use some fairly recent
developments in Several Complex Variables to complete symplectic
jets; a problem that comes up in accelerator physics.
Wed 28 Sept @ VU, 16:00-17:00, Room TBA: Jens Rademacher Title: Unfolding heteroclinic networks of equilibria and periodic orbits with tangencies
Abstract : In this talk I present a method to study bifurcations from
heteroclinic network in ordinary differential equations of arbitrary
dimension. These networks can consist of equilibria and periodic
orbits, and the heteroclinic connections can be tangent. The problem
of bifurcating solutions in a neighborhood of this network is reduced
to algebraic equations by a kind of Lyapunov-Schmidt reduction,
closely related to Lin's method. The abstract result
will be explained through some examples.
Wed 14 Sept @ UvA, 16:00-17:00, Room G0.05: Sara Lombardo (Northumbria University Newcastle and VU)
Title: Three-wave mixing in quadratic nonlinear crystals
Abstract : The resonant interaction of three waves (3WRI) is an important process in physics describing the resonant mixing of three quasi-monochromatic waves in weakly nonlinear and dispersive media. It appears in various physical contexts, notably fluid dynamics and optics. In quadratic nonlinear crystals parametric three-wave mixing provides a versatile means of achieving widely tunable frequency conversion of laser light. In this talk I will consider mathematical features of the 3WRI model, in particular a novel solution describing three locked dark pulses (simulton) will be presented. The dark-dark-dark triplet is found using standard Darboux-Dressing Transformations adapted to construct soliton solutions for a class of integrable PDEs with matrix rather than scalar coefficients. The soliton dynamics associated to this class of PDEs (also know as boomeronic-type equations) may be richer than those of standard solitons.
Wed 18 May, 16:00-17:00, Room P-624: Frederic Bourgeois (Brussels)
Title: S^1-equivariant symplectic homology and families of Hamiltonians
Abstract : This is joint work with Alexandru Oancea. The aim of this talk is to explain the definition of an S^1-equivariant version of symplectic homology, via a Borel-type construction. As an important ingredient of this construction, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.
Tue 3 May, 16:00-17:00, Room M-664: Tanja Eisner (UvA) Note the unusual date.
Title: On rigidity sequences
Abstract: We study rigidity sequences for weakly mixing measure preserving dynamical systems and in particular for linear systems in Banach spaces and show the connections to ergodic theory, number theory and functional analysis. This is a joint work with Sophie Grivaux.
Wed 20 April, 16:00-17:00, Room P-624: Adrian Muntean (Eindhoven)
Title: Homogenization of a Locally-Periodic Medium with Areas of Low and High Diffusivity
Abstract: We aim at understanding reaction and transport in those porous materials that present regions with both high and low diffusivities. For such scenarios, the transport becomes structured (here: micro-macro), while the reaction will be mainly hosted by the micro-structures of the low-diffusivity region. The geometry we have in mind include perforations (pores) arranged in a locally-periodic fashion. We choose a prototypical reaction-diffusion system (of minimal size), discuss its formal homogenization – the heterogenous medium being now assumed to be made of zones with circular areas of low diffusivity of x-varying sizes. We report on two type of results. On one hand, we rely on formal asymptotic homogenization, suitable use of the level sets of the oscillating perforations combined with a boundary unfolding technique to derive the upscaled model equations. On the other hand, we prove the weak solvability of the limit two-scale reaction-diffusion model. A special feature of our analysis is that most of the basic estimates (positivity, boundedness, uniqueness, energy inequality) are obtained in x-dependent Bochner spaces. Finally, the homogenization limit is proven rigorously by means of a suitable corrector estimate (an upper bound on the convergence rate). This is joint work with Tycho van Noorden (University of Erlangen-Nuremberg, Germany).
Wed 6 April, 16:00-17:00, Room P-624: Sebastiaan Janssens (Utrecht)
Title: Discrete-time dynamics on a space of measures
Abstract: Consider a population of agents (e.g. cells or biological oscillators such as integrate-and-fire neurons) whose individual state (i-state) corresponds to a point moving on the one-dimensional unit circle $S$. Assume furthermore that interaction between agents is indirect, via contribution to and dependence on a so-called \emph{environmental condition}. Then the population state (p-state) at a particular moment is given by a measure $\mu \in M_+(S)$, the cone of finite positive Borel measures on $S$. Depending on the presence of birth and death effects, $\mu$ may or may not be a probability measure. We investigate the existence, stability and bifurcation of Dirac-type periodic solutions of the p-state dynamics by studying the fixed points of an associated return map defined on $M_+(S)$. The underlying interpretation is that a stable (unstable) Dirac fixed point corresponds to a synchronised (de-synchronised) p-state.
Wed 9 March, 16:00-17:00, Room P-624: Onno van Gaans (Leiden)
Title: Stationary solutions of stochastic delay differential equations driven by Levy processes
Abstract: Under what types of stochastic perturbations will a stable deterministic delay differential equation remain stable? Stability of stochastic equations is here interpreted as existence of a stationary solution. Several approaches and results will be discussed for stochastic perturbations with non-Lipschitz coefficients and processes with jumps.
Wed. 23 Feb, 16:00-17:00, Room P-624: Jaap Eldering (Utrecht)
Title: The Perron method: what, why, and how to apply it to NHIMs.
Abstract: I will start with a review of the Perron method for proving stable and unstable manifolds of hyperbolic fixed points. This is an alternative to the graph transform method, and I will explain its (dis)advantages. Secondly, I will give an overview of how the Perron method can be generalized to apply it to normally hyperbolic invariant manifolds (NHIMs). This can be done by only linearizing the normal directions.
Wed. 9 Feb, 16:00-17:00, Room P-624: Joost Hulshof (VU)
Energy concentration for the harmonic map heat flow.
Abstract: It is well known that the gradient flow of the Dirichlet energy of maps from the disk to the unit sphere may develop singularities. We present a self-contained approach which, in the radially symmetric case, describes that there is only one well-defined scale in which a bubble appears and bubbles off.

2010

Wed. 20 Oct. 16:00-17:00, Room P-624. Han Peters (UVA)
Title: Fatou components in two complex variables.
Abstract: Roughly speaking, a Fatou component is the largest open connected set for which all orbits behave similarly. As such, Fatou components play an important role in our understanding of holomorphic dynamical systems. For the iteration of polynomials and rational functions in the conplex plane, Fatou components are very well understood. By a result of Sullivan every Fatou component is pre-periodic, and periodic Fatou components are completely classified. In higher dimensions the situation is quite different. Whether every Fatou component is pre-periodic is known only in special cases, and the classification of periodic Fatou components is not nearly complete. In this talk I will describe what is currently known and discuss recent research with M. Lyubich
Wed. 6 Oct. 16:00-17:00, Room M-664. Sander Hille (Leiden)
Title: Reverse engineering of the auxin transport system in Arabidopsis plants.
Abstract: In plants the growth hormone auxin is transported from the top of the plant downwards through the stem. In various experiments in the Plant BioDynamics Lab (PBDL) at Leiden University the dynamical properties of this transport mechanism at macroscopic level are measured for the model plant Arabidopsis thaliana. We will discuss these experimental results and the various ways we needed to model the system mathematically in the form of a system of coupled ordinary and partial differential equations in order to fit the available data. Moreover, we will present how well-designed macroscopic measurements and mathematical modeling together, in close collaboration with the experimental biologists at PBDL, are able to draw conclusions on the structure and functioning of the system at mesoscopic -- cellular -- level.
Wednesday 22 September, 14:00-17:00, Room 08A05 (in the main building of the VU): Mathematics, Science & Engineering. This symposium is organized to bid farewell to Rein van der Hout, whose job at the VU has ended this summer.
Programme:
14:00-14:50 Cas van der Avoort (NXP): Glass ceiling for mechanical micro-resonators
14:50-15:00 Break
15:00-15:50 Michiel Bertsch (IAC, Rome 2):  Heat flow for director fields
15:50-16:00 Break
16:00-16:50: Bert Peletier (emeritus Leiden): Drug delivery to the brain in the presence of high-affinity proteins and lipids
16:50-17:00 Closing
17:00-..... Drinks
Wednesday 23 June, 11:00-12:00, Room: S-664.Martina Chirilus-Bruckner (CWI).
Title: Interaction of Pulses in Nonlinear Wave Equations
Abstract: The interaction of pulses, i.e. spatially and temporarily oscillating waves modulated by a spatially localized envelope, is described via an extended perturbation approach that provides explicit formulas for interaction effects such as a position shift or a shape deformation of the interacting pulses.
The analysis involves a reduction to amplitude equations (based on Fourier or Bloch analysis) and a rigorous justification of these (using energy estimates or semigroup theory). The presented method is applicable to a wide class of nonlinear, dispersive equations in 1+1 dimensions. In particular, it can be carried out for nonlinear wave equations with periodic coefficients which arise, e.g., as model equations for light propagation in photonic crystals.
Wednesday 9 June, 16:00-17:00, Room F-630. Sander Hille (Universiteit Leiden).
This talk has been cancelled.
Tuesday 8 June, 16:00-17:00, Room S-664. Lennaert van Veen (Univ. Ontario Inst. Technology).
Title: The tangled edge of turbulence in bursting Couette flow.
Abstract: In recent years, the scale of dynamical systems-type computations in turbulence research has increased spectacularly. Equilibrium and periodic solutions have been computed for Couette flow, pipe flow and many other geometries. One of the goals of these computations is to explain the process of turbulent bursting in shear flows. Bursting occurs in the presence of an asymptotically stable laminar flow, so that ordinary bifurcation scenarios do not offer an explanation. Instead, the current focus is on "edge states," i.e. saddle-type equilibria or periodic solutions that live on a boundary between turbulent and laminar behaviour. We should be able to clarify the bursting process if we know the geometry of the (un)stable manifolds of such states. However, the systematic computation of these manifolds is a hard task. We present a recently developed algorithm for the computation of unstable manifolds and its application to turbulent Couette flow. This algorithm uses matrix-free linear solving and comes with a strong convergence result. Initial computations indicate that the (un)stable manifolds of an edge state in turbulent Couette flow form a homoclinic tangle, an observation with far-reaching implications for our understanding of the transition to turbulence.
Wednesday 12 May, 16:00-17:00, Room F-630. R. Hindriks (VU). Title: Data-driven dynamical models for spontaneous MEG oscillations
Abstract: Large-scale brain activity as measured with magnetoencephalography (MEG) during rest typically shows oscillatory features. The intrinsic dynamics of these oscillations and the way in which spatially separated oscillations are coordinated is currently not well understood. In this talk I will focus on a recent development within the field of time-series analysis for MEG oscillations, namely the construction and estimation of explicit dynamical models in continuous-time. In particular, we focus on deterministic coupled limit-cycle models in the weak coupling regime, and on two-dimensional Fokker-Planck equations. Both techniques will be applied to MEG data-sets recorded from human subjects.
Wednesday 28 April, 16:00-17:00, Room F-630. G. Pfander (Jacobs University Bremen)
Title: A sampling theory for operators
Abstract: The classical sampling theorem, attributed to Whittaker, Shannon, Nyquist,
and Kotelnikov, states that a bandlimited function can be recovered from its
samples, as long as we use a sufficiently dense sampling grid. Here, we
review the recent development of an operator sampling theory which allows for
a widening of the classical sampling theorem. In this realm, bandlimited
functions are replaced by bandlimited operators, that is, by
pseudodifferential operators which have bandlimited Kohn-Nirenberg symbols.
Similar to the Nyquist sampling density condition alluded to above, we
discuss sufficient and necessary conditions on the bandlimitation of
pseudodifferential operators to ensure that they can be recovered by their
action on a single distribution. In fact, we show that an operator with
Kohn-Nirenberg symbol bandlimited to a Jordan domain of measure less than one
can be recovered through its action on a distribution defined on a
appropriately chosen sampling grid. Further, an operator with bandlimitation
to a Jordan domain of measure larger than one cannot be recovered through its
action on any tempered distribution whatsoever, pointing towards a
fundamental difference to the classical sampling theorem where a large
bandwidth could always be compensated through a sufficiently fine sampling
grid. The dichotomy depending on the size of the bandlimitation is related to
Heisenberg's uncertainty principle.
Wednesday 14 April: no seminar (NDNS+ workshop in Eindhoven)
Wednesday 17 February, 16:00-17:00, Room F-630. Antonios Zagaris (Twente).
Title: Phytoplankton-nutrient dynamics in oligotrophic envirnoments
Abstract: In this talk, we will introduce a system of two coupled reaction-diffusion PDEs modeling phytoplankton-nutrient dynamics in an oceanic environment and for a single phytoplankton species depending on a single nutrient (and light) for its survival. In the
first half of this talk, we will look closely into the linear stability problem for the trivial steady state (no phytoplankton) and use this asymptotic analysis to identify the emergence of localized structures. These turn out to fall into two categories: deep-chlorophyll maxima, in which the plankton concentration is localized in an interior point of the water column, and benthic
layers, in which the plankton concentration is localized in the bottom of the column. This first half will close with a brief ecological interpretation of our findings. In the second half, we will look into the weakly nonlinear stability problem for the bifurcating deep-chlorophyll maxima. The peculiarity of this problem is focused around the existence of an infinite number of latent (non-bifurcating) modes which invariably have to be included in the analysis. Explicit asymptotic results may nevertheless be derived, not only close to the bifurcation point, but in a far bigger regime. In fact, the talk will conclude with the deep-chlorophyll maximum disappearing in a saddle-node bifurcation, thus offering its place to a secondary pattern
Wednesday 3 February, 16:00-17:00, Room F-630. Speaker: Matteo Sommacal (La Sapienza, Rome).
Title: Towards a Theory of Chaos Explained as Travel on Riemann Surfaces
Abstract: Recently, a mechanism to explain the onset of irregular (chaotic) motions in a dynamical system, in terms of the singularity structure of its solutions, was introduced. The dynamics defined by a certain (paradigmatic) set of three coupled (complex) first-order ODEs, featuring two coupling constants, will be illustrated. It is shown that the system under study can be reduced to quadratures which can be expressed in terms of elementary functions. Despite the integrable character of the model, the general solution is a multiple-valued function of time (considered as a complex variable), and we investigate the position and nature of its branch points. For rational values of the coupling constants, the system is isochronous and explicit formulae for the period of the solutions can be given. For irrational values, the motions are confined but feature aperiodic motion. In this case, an argument is introduced to explain why sensitive dependence on initial conditions is expected. The system shows a rich dynamical behaviour that can be understood in quantitative detail since a global description of the Riemann surface associated with the solutions can be achieved. This toy model is meant to provide a paradigmatic first step towards understanding a certain novel kind of chaotic behaviour. This work has been carried out in collaboration with F. Calogero and P. M. Santini of the Università degli Studi di Roma "La Sapienza" (Italy) and D. Gomez-Ullate Oteiza of the Universidad Complutense de Madrid (Spain).

2009

Wednesday 9 December, 16:00-17:00, Room M6.16. Onno Bokhove (Twente).
Title: Variational water wave model with accurate dispersion and vertical vorticity
Abstract: A new water wave model has been derived which is based on variational techniques and combines a depth-averaged vertical (component of) vorticity with depth-dependent potential flow. The model facilitates the further restriction of the vertical profile of the velocity potential to n-th order polynomials or a finite element profile with a small number of elements (say), leading to a framework for efficient modelling of the interaction of steepening and breaking waves near the shore with a large- scale horizontal flow. The equations are derived from a constrained variational formulation which leads to conservation laws for energy, mass, momentum and vertical vorticity. It is shown that the potential flow water wave equations and the shallow-water equations are recovered in the relevant limits.
Wednesday 25 November, 16:00-17:00, Room M6.16. Antonios Zagaris (Twente).
This talk has been cancelled.
Wednesday 11 November, 11:00-12:00, Room P-656. Charlene Kalle (Warwick)
Title: Beta-expansions and invariant measures
Abstract: An expression of the form $x = \sum_{n=1}^{\infty} b_n / \beta^n$ with all numbers b_n in some fixed and finite set of real numbers A is called a \beta-expansion of x with digits in A. A \beta-transformation is a map from an interval to itself that can be used to generate beta-expansions of the numbers in its domain. We study the invariant measure that is absolutely continuous wrt Lebesgue of such a transformation. By constructing a version of the natural extension of the transformation we obtain an expression for the density function of this invariant measure.
Wednesday 28 October, 16:00-17:00, Room M6.16. Marco Mazzucchelli (MPI Leipzig)
Title: The Conley conjecture for Tonelli systems
Abstract: On a closed conﬁguration space, a time-periodic Lagrangian is called Tonelli when its restriction to any ﬁber of T M is superlinear with positive deﬁnite Hessian. In this talk I shall prove that, for any Tonelli Lagrangian with global Euler-Lagrange ﬂow, the associated Euler-Lagrange system admits inﬁnitely many periodic solutions. More precisely, I will show that there are inﬁnitely many contractible periodic orbits with a priori bounded mean action and either inﬁnitely many of them are 1-periodic or their basic period is unbounded. This result conﬁrms the Conley conjecture for Tonelli Hamiltonian systems on the cotangent bundle of closed manifolds
Wednesday 30 September, 16:00-17:00, Room M6.16. Felix Schlenk (Neuchâtel). Title: Product Lagrangian tori in tame symplectic manifolds Abstract: A product torus in standard symplectic space R^2n is the n-fold product of circles in the plane. A product torus in a symplectic manifold is the image of such a torus under a Darboux chart. I'll try to explain how such tori can be classified up to Hamiltonian isotopy. This is work joint with Yura Chekanov.
Wednesday 14 October, 16:00-17:00, Room M6.16. Carlota Cuesta (Nottingham). Title: Front propagation in a heterogeneous Fisher equation and in a
pseudo-parabolic equation Abstract: We address front propagation into unstable states in two evolution PDEs of
the types mentioned in the title. That is the mechanisms by which, under an initial perturbation of an unstable state, stable patterns `win' over the unstable state invading its domain.
The homogeneous Fisher equation has been studied widely in terms of its front propagation properties. The steady state 0 is linearly unstable, whereas 1 is stable. An initial perturbation of 0 grows and propagates with speed that tends to a constant for large time; the solution approximates a travelling wave.
Travelling wave solutions exist for all wave speeds larger than or equal to 2. In the Fisher case it is well-known that for sufficiently rapidly decaying initial data, the front selects the minimal wave speed, whereas for slow decaying initial conditions speeds faster than the minimal are
realised. In this talk I will recall the formal approach that leads to the later results before giving the ones for the heterogeneous Fisher and the pseudo-parabolic equation. These can be summarised as follows: in terms of the front speed selection mechanism, the Fisher equation represents a border-line case of the heterogeneous one in a way that will be discussed. In the pseudo-parabolic equation the decay of the initial condition has no effect in front speed selection.
Wednesday 16 September, 16:00-17:00, Room M6.16. Gaia Lupo (Perugia). Title: Initial/Moving Boundary Value Problems for the Nonlinear Heat Equation Abstract: A nonlinear evolution equation of diffusive type will be analysed as a
mathematical model for heat conduction in high polymer systems and in simple
monoatomic metals of Storm-type. In such a context, a half-line problem with a
prescribed time-dependent heat flux at the origin will be presented and the
construction of the Dirichlet-to-Neumann map on a moving boundary will be
analysed. Some explicit examples will be also considered.
Wednesday 24 June, 14:30-15:30, room P.656: A. Momin (Max Planck Institute, Leipzig)
Intersections and Contact Homology
I will describe a version of cylindrical contact homology on the complement of a collection of closed, elliptic Reeb orbits. The given orbits are used as a barrier to restrict the gradient-like cylinders defining the differential using the phenomena of positive intersections of pseudoholomorphic curves.
Monday 6 April, 16:00-17:00, room P.656: I. Stamova (Bourgas Free University)
Stability analysis of impulsive functional differential systems and applications
Many real world processes and phenomena in nature, science and technology are characterized by the fact that system parameters are subject to short-term perturbations in time. An adequate apparatus for mathematical simulation of such processes and phenomena is the Impulsive Differential Equations. (IDEs) A natural generalization of IDEs is the Impulsive Functional Differential Equations (IDFEs). These equations are an adequate mathematical?model of the processes which are characterized by the change of jumps of their state as well as by the fact that the process under consideration depends on its history at each moment of time. The lecture is dedicated to a development of stability theory for IDFEs. Many applications and some open problems will be discussed.
Wednesday 25 March, 16:00-17:00, room P.656: J. van de Leur (Universiteit Utrecht)
LU- factorization and hierarchies of differential equations
Using the Birkhoff-factorization of Loop groups, i.e. , some kind of LU-factorization, one obtains in an elementary way a hierarchy of PDE's. I will show this for the loop group of type GL(n). The simplest equations in this hierarchy are the n-wave equations.
Wednesday 11 March, 16:00-17:00, room P.656: R. Driesse (UvA)
Bifurcations from robust homoclinic cycles
Thursday 26 February, 16:00-17:00, room S6.48: H. Waalkens (RUG)
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 main approach 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 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. In fact, such a dividing surface is 'spanned' by a normally hyperbolic invariant manifold (NHIM) whose stable and unstable manifolds moreover form the phase space conduits for the reaction. The NHIM can be viewed as the mathematical manifestation of the transition state as an unstable invariant subsystem poised between reactants and products.
In the second part of my talk I will discuss the quantum mechanics of reactions, and the role that the classical phase space structures play for these. This relationship can be studied in terms of a quantum normal form. The two main quantum imprints of the transition state are the quantization of the so-called cumulative reaction probability (the quantum analogue of the classical flux) and quantum resonances which describe the decay of wavepackets initialized on the transition state. The quantum normal form can be formulated as an explicit algorithm which, when implemented on a computer, leads to a very efficient method to compute both cumulative reaction probabilities and quantum resonances. The talk summarizes joint work with Roman Schubert and Stephen Wiggins from Bristol University.
Wednesday 11 February, 16:00-17:00, room C6.48: L. Sella (CWI)
Algorithms for Computation of Symbolic Dynamics
The theory of symbolic dynamics is a powerful tool to study discrete dynamical systems, in particular it is important to analyse qualitative properties of systems which exhibit chaotic dynamics. In the first part of this talk we present algorithms for the computation of symbolic dynamics and computation of topological entropy - a quantity which characterizes the level of chaos of the system - for one dimensional piecewise-continuous maps. We also show how to study the discrete dynamics of an hybrid system by applying results from symbolic dynamics to the return map of the hybrid system. In the second part of the talk we present algorithms for computation of symbolic dynamics and entropy of piecewise-affine two dimensional maps. For this case we show the implementation of a method based on the Conley index of decomposition of disconnected index pair and we compare it with the already well developed approach based on trellis of fixed point tangle. Finally we mention possible extensions of this first technique to higher dimensions.

2008

Wednesday 17 December, 16:00-17:00: Oleg Makarenkov (Imperial College, London)
Topological degree approach to study bifurcations of periodic solutions in perturbed planar systems
Classical conditions for bifurcation of periodic solutions from a cycle x0 in periodically perturbed hamiltonian systems are due to Melnikov [1963]. There were assumed that the cycle x0 is nonsingular and that the corresponding bifurcation function possesses a nonsingular zero θ The singular situation has been considered by Yagasaki [1996] and involves some implicit assumptions. The approach of both authors is based on the Lyapunov-Schmidt reduction. This talk proposes an alternative approach based on the topological degree theory and does not depend on whether the cycle x0 or zero θ is singular or not.
Thursday 23 October, 16:00-17:00, room S2.01: Georg Prokert (Technische Universiteit Eindhoven) Justifying the Thin Film approximation: A rigorous limit result for Stokes flow driven by surface tension In the moving boundary problem of Stokes flow driven by surface tension, we pass to the limit of small layer thickness. In an appropriate scaling, the limit evolution is given by the well-known Thin Film equation. While this is straightforward on the level of formal asymptotics, a rigorous analysis has to deal with the degeneracy of the limit which is reflected e.g. in the fact that a first-order evolution equation is replaced by a limit problem of order four. Our main techniques are uniform energy estimates in appropriately scaled Sobolev norms of sufficiently high order, based on parabolicity. This is joint work with M. Günther, Leipzig.
Wednesday 24 September, 16:00-17:00, room S2.05: Martijn Zaal/Blaz Mramor (VU University Amsterdam)
Linear Stability of Osmotic Cell Swelling
A basic mathematical model for cell swelling by osmosis will be introduced. Some results on the stability of the linearized problem when varying one of the parameters will be presented.

Unbounded non-singular strange attractors
We construct a class of unbounded strange attractors. The construction is based on the so called geometric Lorenz attractor and hints at the existence of unbounded strange attractors, robust under uniform C^2 perturbations.
Wednesday 17 September, 14:30-15:30, room S2.03: Ferdinand Verhulst (Utrecht University)
Resonance and near-resonance in a wave equation
Normal mode manifolds, also called Lyapunov manifolds, arise naturally in ODEs and PDEs. A basic question is then whether these manifolds can be continued for small ε>0. A second basic question is whether the Lyapunov manifolds persist for increasing ε and other changes of relevant parameters. Possible tools to study these questions are averaging-normalization and numerical bifurcation theory. As we will show, the combination of both techniques is very powerful. We will describe two theorems that can be used in an infinite dimensional setting. The technique of averaging-normalization produces a short-cut to normally hyperbolic manifolds that emerge from the normalized equations because of the dominating presence of slow-fast dynamics. We demonstrate this for a parametrically excited wave equation. Our analysis shows that a complicated bifurcational stucture exists for relatively small values of the small parameter ε.
Wednesday 4 June, 16:00-17:00, room S2.01: Alberto Abbondandolo (Università di Pisa)
On the asymptotic Maslov index
The asymptotic Maslov index is a real number which can be attached to an invariant measure of a Hamiltonian dynamical system. We shall discuss its main properties, with special emphasis on systems with compact and simply connected configuration space, where the existence of invariant measures with a prescribed asymptotic Maslov index is easy to establish. We shall argue that the asymptotic Maslov index might play the role of the rotation vector in Mather theory.
Wednesday 21 May, 16:00-17:00, room S2.09: Peter van Heijster (CWI, Amsterdam)
Front and pulse dynamics in a three-component system
We study the dynamics of multi-front solutions of a specific three-component reaction-diffusion system. First, we briefly consider the existence and stability of the stationary patterns -- a 1-pulse or 2-front and a 2-pulse/4-front -- by singular perturbation and Evans functions techniques. Then, we use a renormalization group method to rigorously deduce the system of ODEs that govern the front dynamics. Based on our knowledge of the stationary points of this system, i.e. the stationary patterns, we are able to give an accurate description of the dynamics of N-front patterns (for N not too large).
Wednesday 7 May, 16:00-17:00, room S2.09: Jens Rademacher (CWI, Amsterdam)
The Hyperbolic Continuum Limit of FPU chains: Dispersive and Nonclassical Shocks
Towards the continuum description of FPU mono-atomic chain models in the hyperbolic space-time scaling, we systematically study discrete approximations to Riemann problems composed of constant states by numerical experiments. In the hyperbolic scaling the expected continuum model is a system of conservation laws so that solutions to Riemann problems provide building blocks for a general solution. The naive continuum limit is the p-system of mass and momentum conservation, but the FPU chain also conserves energy, which rules out the Lax-theory for shocks. For non-convex or non-concave flux the classical hyperbolic theory does not even apply to the p-system, and there exist energy conserving shocks. We rigorously study their occurrence and properties in the p-system, and, as a main new result, show that the macroscopic FPU chain can generate supersonic undercompressive shocks. Moreover, we show how non-classical Riemann solvers of the p-system must be adapted to give macroscopic Riemann solvers of the FPU chain in the case of at most one turning point of the flux.
Wednesday 9 April, 16:00-17:00, room S2.01: Jason Frank (CWI Amsterdam)
Statistical accuracy in the numerical discretization of geophysical fluids
In numerical weather prediction and climate research numerical simulations are conducted long beyond the point for which global errors have completely saturated the solution. This begs the question of whether any meaningful conclusions can be drawn from the simulation data. We will address this in an idealized setting by deriving equilibrium statistical mechanics theories for three related discretizations of the quasigeostrophic potential vorticity equation (Arakawa 1966), having mutually distinct conservation properties. Numerical experiments agree with the statistics in each case, indicatinging that statistical mechanics is a useful tool for numerical analysis. On the other hand, both the methods considered and the statistical theories resulting from them are known to fall short of the statistics of the continuum. We discuss why this is and propose an improved discretization.
Friday 14 March, 16:00-17:00, room F6.64: Sara Lombardo (VU University Amsterdam)
Accelerated solitons: from integrable systems to nonlinear optics
Accellerated solitons (also known as Boomerons) are soliton solutions of coupled nonlinear PDEs. Contrary to normal solitons they do not move as free particles but feature instead an accelerated motion. Introduced many years ago, boomerons remained strange waves, rather a mathematical curiosity than of physical interest. It has been recently realised that this is not the case; the breakthrough occurred when it was realised that also the well known 3 Wave Resonant Interaction equations, among other systems, possess boomeronic solutions. This opens the way to applications, particularly in nonlinear optics. In the first part of the talk I will review the state of the art and discuss a few optical processes in quadratic media; in the second part I will discuss solution techniques for a family of PDEs admitting boomerons.
Wednesday 13 February, 16:00-17:00, room S2.03: Michał Wojtylak (VU University Amsterdam)
Krein spaces and definitisable operators
Let us consider a complex Hilbert space (H,<,>) and a bounded, selfadjoint operator J such that J^2=I. It is not hard to see that J is a difference of two orthogonal projections. On the space H we define a new inner product by [f,g]=< J f,g>. Although this inner product is not positive definite, it allows us to define the adjoint of an operator. We will discuss spectral properties of selfadjoint (in the sense of the new inner product) operators. Moreover, we will introduce a class of definitisable operators and provide a version of a spectral theorem for this class. As an example a Sturm-Liouville operator will be considered.

2007

Wednesday 5 December, 13:30-14:30, room F2.53: Florian Wagener (University of Amsterdam)
A stochastic bifurcation theory for discrete time stochastic systems
We propose a bifurcation theory of smooth stochastic dynamical systems that are governed by everywhere positive transition densities. The local dependence structure of the unique strictly stationary evolution of such a system can be expressed by the ratio of joint and marginal probability densities; this 'dependence ratio' is a geometric invariant of the system. By introducing a weak equivalence notion of these dependence ratios we arrive at a bifurcation theory for which in the compact case, the set of stable (non-bifurcating) systems is open and dense.
Wednesday 21 November, 16:00-17:00, room S2.03: Chris Stolk (University of Twente)
Reflection seismology in presence of multipathing
Reflection seismology is a branch of seismolgy that uses reflected seismic waves to produce images of the Earth's subsurface. The method employs sources at the surface, that can emit a short pulse of seismic waves. The reflected signals are recorded using a large number of receivers at the surface. From these data, both the position of the reflectors, and the velocity function between the reflectors and the surface must be estimated. For this, signals from a large number of sources and receivers are combined. In this talk we discuss complications that occur when the waves can travel along different raypaths from a subsurface point to the surface, due to e.g. focussing of waves.
Wednesday 7 November, 16:00-17:00, room S2.01: Leo Maas (NIOZ)
Internal Waves and Wave Attractors
The abstract for this talk can be found here
Wednesday 24 October, 16:00-17:00, room S2.03: Jean-Philippe Lessard (Rutgers University and VU University Amsterdam)
Toward a Computer-assisted Proof of an Old Conjecture in Delay Equations
The scalar delay equation y'(t)=-ay(t-1)[1+y(t)] , (a>0) often called the Wright's equation is arguably one of the most studied equation in the field of nonlinear functional differential delay equations since the mid 50's. A conjecture that was made by Jones in 1962 is that the Wright's equation has a unique slowly oscillating periodic solution (SOPS), for every fixed a>pi/2. Partial progress in trying to prove the conjecture was made by Xie and Nussbaum in the 90's, but they could prove the result only for a>5.67. In this talk, we give a brief history of the problem and show how the use of rigorous numerical continuation techniques can help in getting significant progress toward the proof of the conjecture.
Wednesday 10 October, 16:00-17:00, room S2.03: Martin Bootsma (Utrecht University)
Modeling of the transmission dynamics of Methicillin-resistant Staphylococcus aureus (MRSA)
Antibiotic-resistance is primarily a hospital problem although increasingly, antibiotic-resistant bacteria are present in the animal and human reservoirs. In this talk, I will discuss a model (a branching process) to estimate the spreading capacity of a hospital clone of MRSA and a clone present among pigs and calves farmers. Furthermore, I will discuss a model, which mimics the so-called "Search & Destroy" policy of Dutch hospitals to prevent transmission of MRSA. I will also discuss what type of interventions are effective according to these models and what the potential benefits are of recently developed tests which can detect MRSA in microbiological samples of patients within hours instead of within 3 days.
Wednesday 26 September, 16:00-17:00, room S2.03: Hermen Jan Hupkes (Leiden University)
Invariant Manifolds and Applications for Functional Differential-Algebraic Equations of Mixed Type
Recently, differential equations involving both delayed and advanced arguments have appeared in an increasing number of models, originating from a wide variety of scientific disciplines. We present recent results concerning the existence of center manifolds and the occurrence of Hopf bifurcations for various types of such equations. In particular, we focus on a differential-algebraic equation with mixed arguments that arises from an economic life-cycle model and exhibit, both analytically and numerically, the presence of periodic cycles in the economy under consideration.
Wednesday 12 September, 16:00-17:00, room F1.31: Rein van der Hout (VU University Amsterdam)
Discrete precipitation phenomena in a reaction-diffusion system
Some reaction-diffusion systems exhibit strikingly regular precipitation-patterns, which have first been observed by R.E. Liesegang in 1896. We discuss a mathematical model, due to Keller and Rubinov. This model contains a nonlocal, discontinuous term which makes the analysis rather hard. We show that this model indeed predicts discrete precipitation bands, but many problems remain unsolved. This is joint work with M. Mimura (Meiji University, Tokyo), D. Hilhorst (Paris-Sud) and I. Ohnishi (Hiroshima).
Wednesday 13 June, 16:00-17:00, room S2.03: Peter van der Kamp (La Trobe University, Melbourne) Multi-sums of products
Wednesday 16 May, 16:00-17:00, room S2.01: Konstantin Mischaikow (Rutgers University)
Modeling Transcriptional Control
Wednesday 2 May, 16:00-17:00, room C6.24: Ale Jan Homburg (University of Amsterdam)
Randomly perturbed diffeomorphisms
Using randomly perturbed circle diffeomorphisms as guide, I will review the dynamics and bifurcations of diffeomorphisms with bounded noise. Of particular interest are bifurcations where the support of a stationary measure explodes. I will explain this scenario and discuss quantitative characteristics. This is joint work with Hicham Zmarrou.
Wednesday 18 April, 16:00-17:00, room C6.38: Heinz Hanßmann (Utrecht University)
On the destruction of resonant Lagrangean tori in Hamiltonian systems
Poincaré's fundamental problem of dynamics concerns the behaviour of an integrable Hamiltonian system under a (small) non-integrable perturbation. Under rather weak conditions K(olmogorov)A(rnol'd)M(oser) theory settles this question for the majority of initial values. The perturbed motion is (again) quasi-periodic, the number of frequencies equals the number of degrees of freedom. KAM theory proves such Lagrangean tori to persist provided that the frequencies are bounded away from resonances by means of Diophantine inequalities.
How do Lagrangean tori with resonant frequencies behave under perturbation? We concentrate on a single resonance, whence many n-parameter families of n-tori are expected to be generated by the perturbation; here n+1 is the number of degrees of freedom. For non-degenerate systems we explain the pattern how these families of lower-dimensional tori come into existence, and then discuss what happens in the presence of degeneracies.
Wednesday 7 March, 16:00-17:00, room R2.24: Eric Séré (Université de Paris - Dauphine)
Periodic orbits of singular Hamiltonian systems
In this talk, I will present a joint work with C. Carminati (University of Pisa) and K. Tanaka (Waseda University, Tokyo), JDE 230 (2006), 362-377.
We consider a noncompact hypersurface H in R2N which is the energy level of a singular Hamiltonian of ``strong force" type. Under global geometric assumptions on H, we prove that it carries a closed characteristic, as a consequence of a result by Hofer and Viterbo on the Weinstein conjecture in cotangent bundles of compact manifolds. Our theorem contains, as particular cases, earlier results on the fixed energy problem for singular Lagrangian systems of strong force type.
Wednesday 7 February, 15:30-17:15, room TBA: Otto van Koert (ULB, Brussel)
Introduction to contact homology
In this talk, we will give an informal introduction to contact homology, a powerful invariant of contact manifolds. It is based on studying the dynamics of the Reeb field and can also be used to detect the existence of closed Reeb orbits. It works, roughly speaking, as follows.
We set up an action functional whose critical points are closed Reeb orbits. Then we do Morse homology on this action functional. In other words, we study a suitable gradient flow of the functional. This gives rise to a homology theory which turns out to be an invariant.
Wednesday 24 January, 16:00-17:00, room F3.01: Yves van Gennip (TU Eindhoven)
Blending polymers with mathematics
Pattern formation is a common phenomenon in nature, ranging from the stripes on fishes and patterns in the desert sand to magnetisation of iron and convection in hot water. A prime example of pattern formation is given by diblock copolymers, two mutually repelling polymers chemically bonded together. During my talk we will consider a variational model for diblock copolymer/homopolymer blends. Due to competing influences on different length scales complicated patterns arise on intermediate scales. We will clearly see these competing influences in our model. In one dimension the model is understood and we will see some results in this case. The close relation between this model and a model for lipid bilayers by Peletier-Roeger will be examined and inspired by properties of this latter model we will look for the possibility of 'partial localisation' in the polymer model, i.e. the tendency of the polymers to aggregate on lower dimensional sets, e.g. surfaces in three dimensional space.

2006

Wednesday 13 December, 14:00-15:00, room F6.64: Jens Rademacher (CWI Amsterdam)
The saddle-node of nearly homogeneous wave trains in reaction- diffusion systems
In joint work with Arnd Scheel (University of Minnesota) we study the saddle-node bifurcation of a spatially homogeneous oscillation in a reaction-diffusion system posed on the real line. We use a novel Liapunov-Schmidt reduction applicable to certain singularly perturbed situations and investigate existence and stability of wave trains with large wavelength that accompany the homogeneous oscillation. We find two different scenarios of possible bifurcation diagrams which we refer to as elliptic and hyperbolic. In both cases, we find all bifurcating wave trains and determine their stability on the unbounded real line. We confirm that the accompanying wave trains undergo a saddle-node bifurcation parallel to the saddle-node of the homogeneous oscillation, and we also show that the wave trains necessarily undergo sideband instabilities prior to the saddle-node.
Monday 4 December, 16:00-17:00, room C6.24: Chris Wendl (MIT, Boston)
Holomorphic foliations and Reeb dynamics in dimension 3
The Weinstein conjecture asserts that periodic orbits always exist for a certain class of (Reeb) vector fields on odd-dimensional (contact) manifolds. These vector fields arise naturally, e.g. as Hamiltonian systems restricted to star-shaped energy hypersurfaces in phase space. We will focus on two recent results related to this conjecture in the 3-dimensional case: the theorem of Hofer, Wysocki and Zehnder, that generic star-shaped energy surfaces always admit either 2 or infinitely many periodic orbits, and another due to Abbas, Cieliebak and Hofer, that the conjecture holds whenever the vector field is determined by a so-called "planar" contact form. Both are obtained using some distinctly low-dimensional features of Gromov's theory of pseudoholomorphic curves, which can sometimes be used to construct singular foliations of a contact 3-manifold. These foliations have some remarkable compactness properties, and suggest a program for solving the Weinstein conjecture in dimension 3.
Wednesday 15 November, 14:00-15:00, room H3.58: Guido Carlet (VU Amsterdam)
Integrable systems and Toda hierarchy
Wednesday 1 November, 14:00-15:00, room C6.38: Pieter Eendebak (Utrecht)
Contact structures and projection methods
In my talk I will present to two themes in my disseration: contact structures and projection methods. I will assume the listener has a basic notion of a partial differential equation and knows the basics of differential geometry. In particular the words manifold, vector field, the Lie bracket of vector fields and vector bundles should ring a bell.
Contact structures are used to study the geometry of partial differential equations. Instead of looking at functions and equations we look at integral manifolds and geometric structures. Symmetry methods for (partial) differential equations have been introduces by Sophus Lie in the 19th century. We formulate the symmetry method as a projection method. Then we show that another method, the method of Darboux, is an example of this projection method as well.
Wednesday 18 October, 14:00-15:00, room F6.30: Daan Crommelin (CWI Amsterdam)
Reconstruction of effective stochastic dynamics from data
Construction of stochastic models that describe the effective dynamics of observables of interest is an useful instrument in various fields of application, such as physics, climate science, and finance. I will discuss a new technique for the construction of such effective models from timeseries. The approach centers on the minimization of an object function that measures the difference between the eigenspectrum of the generator of the stochastic process (for example, the Fokker-Planck operator) and a reference eigenspectrum obtained from the data.
Monday 16 October, 16:00-17:00, room F6.54: Klaus Niederkrüger (Université Libre de Bruxelles)
Contact manifolds that are not energy hypersurfaces
The natural paradigm for contact manifolds are certain energy level sets of Hamiltonian systems. Not all contact manifolds are of that type though. We describe a powerful obstruction to realizing a given manifold as such a level set.
Wednesday 4 October, 14:00-15:00, room F6.30: Konstantinos Efstathiou (Groningen)
Fractional monodromy
The monodromy of the Lagrangian fibre bundles that appear in the study of integrable Hamiltonian systems has attracted a lot of attention, since it is the coarsest obstruction to the existence of global action-angle variables and it was not known classically. Recently, a generalization of the standard notion of monodromy, called fractional monodromy, was introduced by Nekhoroshev, Sadovskii and Zhilinskii. Fractional monodromy appears in certain integrable Hamiltonian systems for which the integral map has a family of singular fibers that correspond to hyperbolic periodic orbits with reflection. We describe fractional monodromy in a system in 1:-2 resonance by constructing a concrete basis of the homology group of the fibers and 'passing' part of the basis through the family of singular fibers. In this construction we use extensively the dynamics of the system.