We show that for large coupling delays the synchronizability of delay-coupled networks of identical units relates in a simple way to the spectral properties of the network topology. The master stability function used to determine stability of synchro
nous solutions has a universal structure in the limit of large delay: it is rotationally symmetric around the origin and increases monotonically with the radius in the complex plane. We give details of the proof of this structure and discuss the resulting universal classification of networks with respect to their synchronization properties. We illustrate this classification by means of several prototype network topologies.
We derive rigorous conditions for the synchronization of all-optically coupled lasers. In particular, we elucidate the role of the optical coupling phases for synchronizability by systematically discussing all possible network motifs containing two l
asers with delayed coupling and feedback. Hereby we explain previous experimental findings. Further, we study larger networks and elaborate optimal conditions for chaos synchronization. We show that the relative phases between lasers can be used to optimize the effective coupling matrix.
Stability of synchronization in delay-coupled networks of identical units generally depends in a complicated way on the coupling topology. We show that for large coupling delays synchronizability relates in a simple way to the spectral properties of
the network topology. The master stability function used to determine stability of synchronous solutions has a universal structure in the limit of large delay: it is rotationally symmetric around the origin and increases monotonically with the radius in the complex plane. This allows a universal classification of networks with respect to their synchronization properties and solves the problem of complete synchronization in networks with strongly delayed coupling.
pydelay is a python library which translates a system of delay differential equations into C-code and simulates the code using scipy weave.
We theoretically study chaos synchronization of two lasers which are delay-coupled via an active or a passive relay. While the lasers are synchronized, their dynamics is identical to a single laser with delayed feedback for a passive relay and identi
cal to two delay-coupled lasers for an active relay. Depending on the coupling parameters the system exhibits bubbling, i.e., noise-induced desynchronization, or on-off intermittency. We associate the desynchronization dynamics in the coherence collapse and low frequency fluctuation regimes with the transverse instability of some of the compound cavitys antimodes. Finally, we demonstrate how, by using an active relay, bubbling can be suppressed.
Time-delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states. We present an application of extended time delay autosynchronization introduced by Socolar et al. to an unstable focus. This system repre
sents a generic model of an unstable steady state which can be found for instance in a Hopf bifurcation. In addition to the original controller design, we investigate effects of control loop latency and a bandpass filter on the domain of control. Furthermore, we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase. We present an analysis of the domain of control and support our results by numerical calculations.
We investigate the possibility to suppress noise-induced intensity pulsations (relaxation oscillations) in semiconductor lasers by means of a time-delayed feedback control scheme. This idea is first studied in a generic normal form model, where we de
rive an analytic expression for the mean amplitude of the oscillations and demonstrate that it can be strongly modulated by varying the delay time. We then investigate the control scheme analytically and numerically in a laser model of Lang-Kobayashi type and show that relaxation oscillations excited by noise can be very efficiently suppressed via feedback from a Fabry-Perot resonator.
