Resonance and Synchronization

Dates

31 January and 1 February 2011

Place

University of Groningen, Zernike Campus, Bernoulliborg, Room 5161.0293.

Registration

Attendance of the workshop is open but if you plan to attend we request that you register before 21 January 2011 by sending an e-mail to X.Liu@rug.nl indicating also during which days you will be present.

Speakers

Domien Beersma (RuG), A model of cell-cell interactions in the biological clock of mammals: emergent properties
Benjamin Biemond (TU Eindhoven), Chaotic saddles in systems with friction
Henk Broer (RuG), Resonance and fractal geometry
Konstantinos Efstathiou (RuG), Unstable attractors and heteroclinic cycles in pulse coupled oscillator networks
Kim Gargar (RuG), Entrainment of circadian clocks
Heinz Hanßmann (Utrecht), Resonance phenomena in Hamiltonian systems
Igor Hoveijn (RuG), Pacer cells and forced coupled oscillators
Martha Merrow (RuG), Coupled oscillators: examples and challenges from circadian biological clocks
Henk Nijmeijer, (TU Eindhoven), Huijgens' synchronisation in mechanical systems
Ferdinand Verhulst (Utrecht), Whitney's umbrella for explaining instability
Nathan van de Wouw (TU Eindhoven), Controlled synchronization via nonlinear integral coupling

Organizers

Henk Broer (RuG)
Konstantinos Efstathiou (RuG)
Xia Liu (RuG)

Program

Monday 31 January 2011

Tuesday 1 February 2011

Abstracts

Domien Beersma (RuG), A model of cell-cell interactions in the biological clock of mammals: emergent properties

Cells in the biological clock of mammals show intervals of electrical activity and rest which alternate with a period of roughly 24 hours. Even cells in isolation show such rhythmicity, but the period varies in a wider interval around 24h. The difference demonstrates that the cells in the tissue interact. To model biological clock behavior we assume that the intervals of rest and activity show normal distributions, both with respect to the variability among cells and to day to day variations within cells. We add simple assumptions about cell-cell interactions and about responses to external light. Simulations show that the integrated system can create synchrony between cells; that the distribution of activity of the cells can adjust to seasonal changes in the photoperiod, and that the system (after being entrained to light-dark cycles of different duration, like 23 or 25 hours) shows a kind of memory of these cycles while single cells do not have that capacity.

Benjamin Biemond (TU Eindhoven), Chaotic saddles in systems with friction

Friction is a pervasive physical effect, which introduces unexpected behaviour in dynamical systems. In systems with friction a set of trajectories with distinct initial conditions can collapse onto a single point. To model this behaviour, differential equations are extended to have set-valued righthand sides. We show that the existence of transversal homoclinic orbits generates a novel type of chaotic saddle in these systems. The dynamical collapse of trajectories implies that the unstable set does not have a fractal structure.

Henk Broer (RuG), Resonance and fractal geometry

The lecture concerns a number of resonance phenomena and the theory behind this in terms of dynamical systems depending on parameters. Apart from two concrete cases, one of which goes back on Christiaan Huygens and the other on the Botafumeiro in Santiago de Compostela, also a number of related `universal' problems will be dealt with that all have to do with Hopf bifurcations in a general sense. It turns out that resonance in the parameter space is organized according to `tongues' the precise form of which can be revealed by Singularity Theory. The corresponding dynamics largely is periodic. In between the tongues sits a fractal set characterized by Cantor sets of positive measure. The corresponding dynamics partly is quasi-periodic. Also chaotic dynamics may occur.

Konstantinos Efstathiou (RuG), Unstable attractors and heteroclinic cycles in pulse coupled oscillator networks

We consider networks of pulse coupled oscillators with non-zero delay. The coupling between the oscillators is given by the Mirollo-Strogatz function. Such networks are used for modelling, for example, the activity in biological neuron networks or the synchronization processes in networks of interacting agents. Because of the non-zero delay the state space of such systems is infinite dimensional. We introduce a metric in the state space in order to study the question of existence of unstable attractors, i.e., of saddle periodic orbits whose stable set has non-empty interior. We prove that for any number of oscillators n >= 3 there is an open parameter region in which the system has unstable attractors. Moreover, in the case of n=4 oscillators we show that there exist unstable attractors with heteroclinic cycles between them.

Kim Gargar (RuG), Entrainment of circadian clocks

The suprachiasmatic nucleus (SCN) is considered to be the seat of the mammalian circadian clock. It has been modeled as a collection of globally coupled N pacers (phase oscillators) each with a constant angular velocity and two alternating states: active and inactive. Each pacer responds to a stimulus at the end of the active state by an instantaneous phase delay and near the end of the inactive state by an instantaneous phase advance whose amounts are proportional to the strength of the stimulus. Entrainment of a pacer to a periodic stimulus is characterized by the stimulus parameters and four pacer parameters: its period, the duration of its active state, and the proportionality constants related to phase delay and phase advance. Numerical results and their biological relevance will be discussed.

Heinz Hanßmann (Utrecht), Resonance phenomena in Hamiltonian systems

Dynamical instability in Hamiltonian systems is often related to occurring resonances. Examples include bifurcations of periodic orbits which are triggered by resonant Floquet multipliers and Arnol'd diffusion which is made possible by the destruction of resonant Lagrangean tori. I will survey the various possibilities and address open problems.

Igor Hoveijn (RuG), Pacer cells and forced coupled oscillators

A collection of neurons situated near the optical nerves is believed to act as a biological clock regulating activity and inactivity in mammmals. This clock is stimulated by the natural 24-hour light-dark cycle, the Zeitgeber. In the literature there exist a vast number of models for this biological clock, ranging from detailed models about bio-chemical processes at cellular level, to models based on the response of mammals to stimuli of light and dark sequences. I will give a few examples showing that almost all models boil down to a system of forced coupled oscillators. Therefore, for all these models one is confronted with the problem of synchronization in such a system. That is, assuming the forcing is periodic, under what circumstances are all oscillators, oscillating with the same period as the forcing? This is not an easy problem. In a simpler setting, for a model of a so-called single phase oscillator, I will give results about synchronization to a periodic Zeitgeber.

Martha Merrow (RuG), Coupled oscillators: examples and challenges from circadian biological clocks

The daily (circadian) biological clock rules our lives, controlling processes from gene expression to physiology and behaviour. It is often stated that the clock is generated from a transcriptional feedback loop but it is built upon various levels of oscillators, e.g., molecules, cells, organs, organism, in a biological construction reminiscent of Mandelbrot Fractals. Even at the molecular level, genetically defined oscillators couple together to make a cellular clock. Cell clocks must find a way to synchronise within their local enviroment (organ clocks) and these masses of cells will establish a stable relationship within the organism. The ‘problem’ of the circadian clock, then, is a one of synchronisation. I will discuss some fundamental questions concerning synchronisation at the different levels and also experimental systems and data that demonstrate the principles of biological (circadian) entrainment from molecules to humans in everyday life.

Henk Nijmeijer (TU Eindhoven), Huijgens' synchronisation in mechanical systems

The purpose of the talk is to review a number of synchronization/coordiantion examples in mechanical systems. This will include the famous Huijgens' pendulum clocks experiment, but also similar results will be discussed for other types of electro-mechanical oscillators. Experimental lab results will highlight some of the results.

Ferdinand Verhulst (Utrecht), Whitney's umbrella for explaining instability

The paradox of destabilization of a conservative system by small dissipation is called in engineering `Ziegler's paradox'. The first complete analytic and geometrical explanation was given by Bottema in 1956. Hoveijn and Ruijgrok succesfully tied in these phenomena with Whitney's umbrella in 1995. We will discuss various examples illustrating the theory, in particular a simple-looking model studied by L.E.J. Brouwer showing various counter-intuitive phenomena. Extensions of the model in the context of gyroscopic stability theory produce some interesting results.

Nathan van de Wouw (TU Eindhoven), Controlled synchronization via nonlinear integral coupling

In this talk we consider the problems of controlled synchronization and regulation of oscillatory systems. For a specific class of nonlinear systems, namely for minimum phase systems with relative degree one, we propose a systematic design procedure for finding nonlinear couplings between the systems - both unidirectional and bidirectional - that guarantee asymptotic synchronization of the systems’ states for arbitrary initial conditions. The corresponding coupling has the form of an integral and it can be considered as a generalized distance between the outputs of the coupled systems. It combines both the low- and the high-gain coupling design in one nonlinear function. The results are illustrated with simulations of coupled Hindmarsh-Rose neuron oscillators.

Participants