Scientific report

International Workshop On State-Dependent Delay Equations
Max Planck Institute for Physics of Complex Systems, Dresden, 12 - 16 October 2009
Scientific Coordinators: Jayme De Luca (Universidade Federal de São Carlos, Brazil), Antony R. Humphries (McGill University, Montreal, Canada), John Mallet-Paret (Brown University, Providence, USA)

This workshop surveyed recent advances and ongoing research across the field of Delay differential equations (DDEs), with particular emphasis on state dependent DDEs, neutral DDEs, mixed-type functional differential equations with both advanced and retarded arguments, and problems with a differential-algebraic structure, including the equation of motion of a charged particle in the action-at-a-distance electrodynamics. State dependent delay equations arise in many applications, but fall outside the scope of the rapidly maturing theory of fixed delay equations, 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.

The workshop brought together researchers from different fields including physics, engineering, physiology and mathematics to exchange recent results, ideas, techniques and problems, and to discuss future directions in the field. Many of the participants were meeting each other for the first time, a clear sign that we were succeeding in the bringing practitioners from different fields together, as evidenced by direct feedback and the article at http://www.dynamicalsystems.org/ma/ma/display?item=315

Even the basic theory of state-dependent delays still raises many open questions, and the analysis and numerical analysis of these problems constituted the main themes of the workshop, and were the common point that united the participants from disparate fields. In his colloquium talk Hans-Otto Walther presented the mathematical framework for considering state-dependent delay equations as dynamical systems with semi-flows on an infinite-dimensional functional space, and the results and open problems of this approach.

Four talks, delivered by Gernot Bauer, Dirk Deckert and C.K. Raju and Savio Rodrigues, considered the equations of motion of charged particles in the action-at-a-distance electrodynamics, which are a neutral mixed-type implicitly state-dependent differential equation, and their formulation and numerical solution as (well-posed) initial value or boundary value problems. These were perhaps the most challenging equations considered during the workshop, but combinations of the different difficulties they pose (including neutrality, advanced and retarded arguments, implicit state-dependency and numerical solution) arose in other equations presented throughout the week.

Mike Mackey showed how delays arise in Bacterial operon dynamics, and other talks considered delays in microvascular blood flow, neural dynamics, epidemiology and population dynamics. State dependent equations (including implicitly state-dependent) were shown to arise naturally in a series of interesting engineering problems, including regenerative cutting and hybrid systems testing.

Analysis and numerical analysis were shown to work in consort in the talk of Alfredo Bellen concerning termination and regularization of solutions of state dependent neutral delay equations. Other numerical analysis talks tackled problems including computation of Lyapunov exponents, boundary value problems, spectral methods and delay dependent stability of numerical methods, the last of which resonated with the analysis talk of Ferenc Hartung considering linearized stability in state-dependent DDEs. Other analysis talks considered singular perturbed problems and slowly oscillating periodic solutions, computer assisted proofs, smoothness of manifolds and co-existence of periodic solutions. If delay problems are hard because of the infinite dimensionality, as are PDEs, partial differential delay equations are doubly hard and Wolfgang Ruess closed the meeting by presenting the theory and open problems of PDDEs.

A particular aim of the workshop was to give young scientists a forum to present their work, and interact with established researchers in the field, and of the 48 participants, 11 were graduate students, and another 11 postdoctoral researchers. These junior scientists participated fully in the workshop with 12 presenting posters in the poster session, and 9 giving talks. Some of them will likely be working in this field for a long time to come.

This was to the best of our knowledge the first workshop dedicated to state-dependent delay equations, and brought together all manner of scientists that had not interacted before. Many interesting discussions ensued, a number of participants indicated that they are in the early stages of exploring completely new collaborations as a result of the workshop. No doubt the fruitions of these collaborations will be revealed at future SDDE workshops, but the first such workshop was a MPIPKS in Dresden, and it was a great success.

For further information please e-mail to: sdde09@pks.mpg.de