The influence of delayed coupling and feedback on dynamical systems has been investigated for several decades. However, the early studies were mostly restricted to a few systems, and powerful mathematical and computational tools to treat these systems were lacking. In recent years, more and more systems are being recognized to be influenced by or to be describable via delayed coupling. Moreover, the application of delay systems to a spectrum of applications is being pursued, ranging from encrypted communication to system control. An important aspect is that now tools have become available that allow one to study delay systems experimentally, via numerical simulations and via analytic treatment in an unprecedentedly detailed and, even more importantly, coherent manner. Therefore, in recent years, dynamical systems with delays have evolved as a major topic in nonlinear sciences. Time delays arise naturally in complex dynamical systems from many different areas of physics, biology, medicine and technology, such as nonlinear optics, electronic circuits, neuroscience, social networks, communication and logistic networks. For instance, delays may occur as memory and latency effects due to finite signal propagation and processing speeds, or may be introduced through closed-loop control schemes. In this Theme Issue, we bring together a selection of leading scientists from different fields working on delay-related problems. The aim is to address fundamental developments and state-of-the-art applications. Special attention is paid to recent foci of interest like, for instance, heterogeneity and robustness of delay-coupled networks, and distributed or statedependent delays. Fundamentally new trends such as information processing in delayed systems and delay control of quantum systems will be included.
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