Handling editor: Hinke Osinga

International Workshops on Delayed Complex Systems October 5-9, 2009 Organized by: Wolfram Just Axel Pelster Michael Schanz Eckehard Schöll   State-Dependent Delay Equations October 12-16, 2009 Organized by: Jayme De Luca Antony R. Humphries John Mallet-Paret

The Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, hosted two workshops in October with a focus on delay equations. The workshop on Delayed Complex Systems was organized by Wolfram Just (Queen Mary Universit of London), Axel Pelster (Universität Duisburg-Essen), Michael Schanz (Universität Stuttgart) and Eckehard Schöll (Technische Universität Berlin) with local support from Renate Seidel and Sofja Roscina. The workshop on State-Dependent Delay Equations was held the following week and was organized by Jayme De Luca (Universidade Federal de São Carlos, Brazil), Tony Humphries (McGill University, Montreal) and John Mallet-Paret (Brown University, Providence) with local support from Katrin Lantsch.

The international workshop on Delayed Complex Systems addressed state-of-the-art applications of delay differential equations in various scientific disciplines. In particular, its aim was to review the systematic investigations of nonlinear dynamical systems with time delay for mechanical, electro- and acousto-optical as well as optically bistable systems and lasers. The meeting also encompassed applications of time delays in neuroscience, where memory effects are important, economy, sociology, and ecology, where the finite propagation time of information must often be considered. Furthermore, time delays have become an important tool for controlling nonlinear dynamical systems, as exemplified by new approaches towards controlling deterministic chaos or noise-induced oscillations and patterns. Complementary to these applications, participants presented fundamental developments in novel theoretical concepts and sophisticated numerical analysis of these systems. Particularly of interest were new insights in the mathematical properties of deterministic and stochastic delay differential equations, including spatially inhomogeneous systems.

Group photo of the workshop on Delayed Complex Systems.

This first workshop was the larger of the two, with about 80 participants. It had a clear focus on the physical sciences aspects of delays, with a healthy contribution from applied mathematics. Synchronization was a theme throughout, with applications in biological as well as physical systems. When it comes to fields of applications, then lasers dynamics was the most prominent. Personal highlights for me were presentations that combined several aspect of the meeting. In this regard I would like to mention the talks by Raj Roy on sensor networks realized by simple laser devices with opto-electronic feedback, by Andreas Amann on time-delayed multi-mode laser networks, and the evening lecture by Ingo Fischer on lasers coupled via relay elements. The contributed presentations were also very good and provided an excellent insight into the types of problems people are currently working on; here recurrent themes were delayed control schemes and stochastic effects. It was also great that two participants of the second workshop talked on state-dependent delay problems in the first week, hence, demonstrating the cohesion between the two events. The mention of an evening lecture indicates that the programme was quite dense. However, there was sufficent time during regular coffee breaks, during the poster session (helped by German beer and wine) and — last but not least — the excursion on Wednesday afternoon. The complex system called the weather showed its friendly side and we were treated to a lush afternoon (20º Celsius!) in Dresden and in the summer residence of Pillnitz, which was reached by Elbe boat.

Group photo of the workshop on State-Dependent Delay Equations.

The international workshop on State-Dependent Delay Equations addressed the fact that physical and biological applications often demand the use of non-constant or state-dependent delays, and realistic models give rise to mixed-type funtional differential equations with both advanced and retarded arguments (for example, travelling waves in nonlinear lattices). Neutral mixed-type implicitly-state-dependent equations fall outside the scope of the rapidly maturing theory of fixed delay equations treated as dynamical systems with semi-flows on an infinite-dimensional functional space, and give rise to challenging problems in both the mathematical analysis of the equations and the numerical computation and analysis of solutions. Although progress has been made in recent years on some model state-dependent problems, in particular monotone problems with positive or negative feedback, the behavior of more general and realistic systems remains poorly understood.

This second workshop was more mathematical in nature, owing to the fact that even the basic theory of state-dependent delays still raises many open questions. The exposition of the theory, for example in terms of the existence of special solutions, was well complimented by contributions on numerical methods. Furthermore, the organisers did a great job of ensuring participation from `interested parties' (which included myself) with quite a number of talks on where state-dependent and advanced-delayed equations arise in applications. It was great to see some colleagues that I had not met for a long time (especially Shui-Nee Chow whom I know from my PhD time!) but, compared to the first week, I knew far fewer participants. This was apparently also the case for other participants, and it shows that the organisers did really manage to bring different communities together. The breadth of the workshop was also reflected in the methods of delivery: the whole range was covered, from blackboard talks to computer animations. As was pointed out, this was the first workshop on delay equations with the word `state-dependent' in the title. And it must be said that the weather appeared to be in quite a state as well. The excursion through Dresden on Wednesday afternoon could not have been more different from the week before: it started to snow! The local organisers had wisely chosen a programme with indoor features. The highlight was the demonstration of the famous Sibermann organ in the Katholische Hofkirche (Catholic Court Church), whose construction was finished in 1755. Taken together, the two workshops were an excellent opportunity to catch up on the latest developments in theory, numerics and applications of systems with delay. Bernd Krauskopf University of Bristol