SCHEDULE OF TALKS

  8:55 - 9:00     Opening

  9:00 - 9:45      J. Jost, Structure formation in systems coupled with transmission delays
  9:45 - 10:30    M. Rosenblum, Delayed feedback suppression of collective rhythmic activity in a neuronal ensemble
10:30 - 11:05    F.M. Atay, Oscillator death in networks of delay differential equations

11:05 - 11:30    coffee break

11:30 - 12:15    M.C. Mackey, Density evolution in systems with delayed dynamics: An open problem

12:15 - 13:30    lunch

14:30 - 15:15    H.-O. Walther, State-dependent delays, linearization, and periodic solutions
15:15 - 16:00    T. Erneux, Delay differential equations modeling physical systems

16:00 - 16:05    short break

16:05 - 16:25    F. Crauste, Bifurcation in a mathematical model of blood production
16:25 - 16:45    D. Savic, Hypothalamo-pituitary-adrenocortical system: A DDE model and its stability
16:45 - 17:05    P. Ranjan, Period doubling and slowly oscillating periodic orbits in DDE models of Internet protocol dynamics

17:05 - 17:30    tea & coffee



ABSTRACTS

Structure formation in systems coupled with transmission delays
Jürgen Jost (Max Planck Institute for Mathematics in the Sciences, Leipzig)

We investigate chaotic oscillators coupled in a network with transmission delays. Surprisingly, the delays facilitate the synchronization of such networks. Moreover, we can see emergent behavior on a longer temporal scale. On this basis, we formulate a concept of emergence through the reduction of the individual degrees of freedom of elements in a system and the transformation of the resulting complexity gain to a larger scale.


Delayed feedback suppression of collective rhythmic activity in a neuronal ensemble
Michael Rosenblum (Potsdam University)

We propose the delayed feedback approach to suppression of collective synchrony in a population of globally or randomly coupled neurons. The method is based on the time-delayed feedback via the mean field. Having in mind possible applications for suppression of pathological rhythms in neural ensembles, we present numerical results for different models of spiking or bursting neurons, including a realistic model of synaptically coupled population of inhibitory and excitatory neurons. Next, we consider the main factors of imperfection of the control scheme and their influence on the suppression efficiency. A theory of the technique is developed, based on the consideration of the synchronization transition as a Hopf bifurcation.


Oscillator death in networks of delay differential equations
Fatihcan M. Atay (Max Planck Institute for Mathematics in the Sciences, Leipzig)

A coupled network of dynamical systems can exhibit a range of interesting behavior that may be qualitatively very different from their behavior in isolation. The so-called oscillator death is one such example, where limit-cycle oscillators stop oscillating when coupled and tend to a stable equilibrium instead. Furthermore, for a network of identical oscillators, such behavior is only possible in the presence of delays. This talk will present a general characterization of oscillator death in the setting of coupled systems of delay differential equations whose oscillatory character results from a supercritical Hopf bifurcation. Using center manifold techniques and ideas from graph theory, the stability of the equilibrium solution is determined in terms of the delays and the spectrum of the graph Laplacian, for both directed and undirected networks. The results are also extended to discrete-time systems, and a complete characterization is given for conditions under which a network of coupled (and possibly chaotic) maps can be driven to a fixed point by an appropriate choice of delays.


Density evolution in systems with delayed dynamics: An open problem
Michael C. Mackey (McGill University, Montreal)

In situations where the dynamics of individual units are governed by ordinary differential equations, the evolution of an initial density of these units is governed by the Liouville equation. Similarly if the dynamics are governed by stochastic differential equations, then the density evolution is determined by the Fokker Planck equation. However, when dynamics are governed by a differential delay equation, there is no known evolution equation for density--indeed it is even unclear what the term `density' means. This talk will outline this problem in some depth, and some of the attempts that have been made to solve it. Most of these attempts have been futile, however, and though this is an important problem for certain areas of application no conclusive results will be presented (unfortunately). Rather, it is my hope to stimulate others to think about ways in which this problem can be tackled.


State-dependent delays, linearization, and periodic solutions
Hans-Otto Walther (University of Giessen)

For differential equations with state-dependent delays, the initial value problem with data in open subsets of the familiar state spaces C([-r; 0];R^n) or C1([-r; 0];R^n) is in general not well-posed. Under mild conditions on the equation, however, there is a submanifold of C1([-r; 0];R^n) on which the IVP generates a semiflow F with continuously differentiable solution operators F(t;.) and further smoothness properties. This yields local invariant manifolds at stationary points and a convenient Principle of Linearized Stability, among others. The ad-hoc technique of freezing the delay and then linearizing the resulting equation with invariant delay, which has been successfully used by several authors for results related to linearization, is explained within the new framework. As applications a Hopf bifurcation theorem for differential equations with state-dependent delay, due to Markus Eichmann, and a global result about existence of stable periodic orbits are presented, the latter for a system describing position control by echo.


Delay differential equations modeling physical systems
Thomas Erneux (Université Libre de Bruxelles)

Delay differential equations modeling physical systems have been formulated for more than a century. For some problems, the delay naturally appears in the system and it has destabilizing effect. This is the case of lasers subject to optical feedback or machining operations experiencing tool chatter. More recently, physicists and engineers have deliberately introduced a delayed feedback in their systems in order to gain stability. This goes from the stabilization of unstable orbits in driven oscillators to sway reduction in container cranes. Three recently studied problems will be considered. The first problem concerns the control of container cranes with a delayed feedback. The second problem deals with sleep disorders and the effect of muscle reflex. Last, we concentrate on experiments using a semiconductor laser subject to incoherent feedback.


Bifurcation in a mathematical model of blood production - Applications to chronic myelogenous leukemia
Fabien Crauste (Université de Pau et des Pays de l`Adour)

We analyze a mathematical model of blood cells production in the bone marrow (hematopoiesis), based on early works by Mackey and Pujo-Menjouet & Mackey. The model is a system of two age-structured partial differential equations. Integrating these equations over the age, we obtain a system of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration. The system describes the evolution of the total cell populations. By constructing a Lyapunov functional, it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium. It is also shown that the unique nontrivial positive equilibrium, the most biologically meaningful one, can become unstable via a Hopf bifurcation. Numerical simulations are carried out to illustrate the analytical results. The study maybe helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases. In particularly, we apply our model to the case of chronic myelogenous leukemia, a cancer of white blood cells for which oscillations have been observed in patients with very long periods (ranging from 30 to 110 days, with an average of 70-80 days) compared to the cell cycle duration.


Hypothalamo-pituitary-adrenocortical system: A delay differential equation model and its stability
Danka Savic (VINCA Institute of Nuclear Sciences)

Hypothalamo-pituitary-adrenocortical (HPA) axis is a system generating a cascade of hormones (cortisol as the end product) with a self-regulatory loop, mostly known for its role in stress response. But, it has another important function: conveying circadian rhythmic signals from the major pacemaker (suprachiazmatic nucleus, SCN) to the periphery, thus according bodily systems with environmental changes. If this function were disturbed, the synchronization of many vital processes would be disrupted and the effects could even be lethal. This is why the stability of the HPA axis dynamics is so important. Are the daily oscillations observed in HPA axis functioning the result of the system’s response to the external pacemaker (SCN) solely or a superposition of external and HPA’s intrinsic time rhythmicity? In general, a reaction system with a negative feedback loop is capable of generating oscillations. If this were the case, the emergence of a sequence of bifurcations leading to multi-dimensional oscillations and eventually to chaotic dynamics would be possible. The empirical data available is not sufficient to suggest that any of these more complicated dynamic patterns are the basis of the observed daily variations of the HPA hormones. The smaller scale peaks superimposed on the diurnal pattern of hormone secretion do not indicate chaos. They reflect the activity of internal ultradian pacemakers and probably stochastic influence of many neurohormones and transmitters that are known to modulate HPA axis functioning. A qualitative mathematical model of the HPA axis dynamics, as a negative feedback mechanism, is constructed and examined using linear stability analysis and Rouche's theorem. It consists of three nonlinear delay differential equations (DDE's) where all the terms are physico-chemically interpretable. The analysis shows that the model is asymptotically stable for realistic values of parameters, i. e. it does not generate circadian oscillations, but only responds to the external pacemaker. The fact that circadian frequency is maintained in the majority of stress-induced disorders (although the hormone amplitude can be increased), meaning that even large perturbations do not change time pattern of the HPA axis activity, supports this theoretical finding and confirms the significance of HPA's role in preserving circadian rhythmicity.


Period doubling and slowly oscillating periodic orbits (SOP) in delay-differential models of Internet protocol dynamics
Priya Ranjan (University of Maryland, College Park)

In this work a novel nonlinear invariant theory based (in)stability results for optimal rate allocation schemes will be presented. We will characterize the global stability conditions with an arbitrary fixed communication delay in the context of optimization framework for the rate allocation problem proposed by Kelly. New results about this system behavior after it loses stability will also be presented using the theory of slowly oscillating periodic orbits and their stability using nonlinear analysis. Various generalizations like presence of time varying delay, distributed delay, feedback based on averaging, stochastic stability will be derived in the invariance framework along with illustration of smooth and non-smooth bifurcations and Slowly Oscillating Periodic Orbits (SOP). We will also explore the applications and roles played by the delay in the feedback control systems like the Internet and Networked Control systems. The role of proposed invariance based-framework as unified approach to treat delay problems in engineering will be emphasized.