Delayed complex systems: an overview.

There does not exist a generally accepted definition for the notion of complex systems in science, but it is a common belief that complex systems show features that cannot be explained by just looking at their constituents. Thus, a complex system normally involves interaction of subunits, and depending on the time scales, the propagation speed of information may become relevant for the dynamics. Therefore, it has been recognized nowadays that time delay may play a vital role in understanding complex behaviour. The new era of complexity science faces two challenges, namely dealing with the non-trivial topology of interacting subunits and the challenge caused by time delay, with the associated dynamics taking place in infinite-dimensional phase spaces. In the past, the subject of time-delay dynamics has been considered as a specialist topic so that only relatively few researchers were attracted. Within the last 15 years, research activities have considerably increased as many new applications have emerged in different areas, such as electronic engineering, controlling chaos, laser physics or neuroscience. Dynamics with time delay is going to play a vital role in new emerging fields of science and technology, for instance, because the speed of modern data processing does not allow finite propagation times of signals to be neglected any more. In addition, there are striking analogies between lasers and neural systems, in both of which delay effects are abundant. It is an intriguing feature of time delay that the phase space of any ordinary differential equation subjected to delay becomes infinite-dimensional. Hence, even simple equations of motion are able to produce extremely complex behaviour and bifurcation scenarios. Time delay has two complementary, counterintuitive and almost contradicting facets. On the one hand, delay is able to induce instabilities, bifurcations of periodic and more complicated orbits, multi-stability and chaotic motion. On the other hand, delay can suppress instabilities, stabilize unstable stationary or periodic states and may control complex chaotic dynamics. The recently held workshop on Delayed complex systems—from 5–9 October at the Max Planck Institute for the Physics of Complex Systems, Dresden, Germany—provided a forum for such topics. We took this opportunity to assemble a list of world-leading experts that now enables us to present an overview of the state of art in this field. The theme issue covers both applications and experiments, as well as mathematical foundations. The individual contributions summarize recent research results, but also address a broader context. Thus, the presentation is kept accessible for a large audience. The 12 articles cover all aspects of delay dynamics, ranging from control and stability, synchronization and numerical approximations, with applications in laser systems, neural science and engineering.