Chaos synchronization in networks of delay-coupled lasers: Role of the coupling phases

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 lasers 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.Comment: 21 pages, 10 figure

Symmetry-breaking transitions in networks of nonlinear circuit elements

We investigate a nonlinear circuit consisting of N tunnel diodes in series, which shows close similarities to a semiconductor superlattice or to a neural network. Each tunnel diode is modeled by a three-variable FitzHugh-Nagumo-like system. The tunnel diodes are coupled globally through a load resistor. We find complex bifurcation scenarios with symmetry-breaking transitions that generate multiple fixed points off the synchronization manifold. We show that multiply degenerate zero-eigenvalue bifurcations occur, which lead to multistable current branches, and that these bifurcations are also degenerate with a Hopf bifurcation. These predicted scenarios of multiple branches and degenerate bifurcations are also found experimentally.Comment: 32 pages, 11 figures, 7 movies available as ancillary file

Neural forecasting: Introduction and literature overview

Neural network based forecasting methods have become ubiquitous in large-scale industrial forecasting applications over the last years. As the prevalence of neural network based solutions among the best entries in the recent M4 competition shows, the recent popularity of neural forecasting methods is not limited to industry and has also reached academia. This article aims at providing an introduction and an overview of some of the advances that have permitted the resurgence of neural networks in machine learning. Building on these foundations, the article then gives an overview of the recent literature on neural networks for forecasting and applications.Comment: 66 pages, 5 figure

Komplexe zeitverzögerte Systeme und Anwendungen auf Laser

In der vorliegenden Arbeit untersuche ich den Einfluss von Zeitverzögerungen in dynamischen Systemen und die Anwendungen auf Laser. Ich konzentriere mich auf zwei Aspekte: (i) die nichtinvasive Kontrolle von periodischen Orbits durch zeitverzögerte Rückkopplung und hier im Besonderen die Stabilisierung von odd-number Orbits und (ii) die Synchronisation von zeitverzögert gekoppelten Systemen. In beiden Fällen besteht die zentrale Frage in der Stabilität von Lösungen unter dem Einfluss der Retardierung. Zeitverzögerte Rückkopplungskontrolle wurde von Pyragas eingeführt um instabile periodische Orbits zu stabilisieren. Es wurde lange Zeit geglaubt, dass sogenannte odd-number Orbits, d.h. Orbits mit einer ungeraden Anzahl von instabilen Floquetmultiplikatoren, mit der Methode nicht zu stabilisieren seien. Dieser Irrglaube wurde kürzlich widerlegt. In der vorliegenden Arbeit analysiere ich detailliert das Gegenbeispiel. Weiterhin konstruiere ich neuartige Rückkopplungsschemata, die erfolgreich odd-number Orbits stabilisieren können und direkt auf experimentelle Situationen anwendbar sind. Weiterhin zeige ich mit Hilfe von Normalformanalysen und numerischen Simulationen die Anwendungen dieser Kontrollmethoden auf Laser. In zeitverzögert gekoppelten Systemen untersuche ich Synchronizationsphänomene und hierbei im Besonderen Chaossynchronisation. Das wichtigste Resultat in diesem Teil betrifft die "master stability function" für allgemeine Netzwerke mit delay-Kopplung. Hier zeige ich, dass die "master stability function" für grosse Verzögerungszeiten eine einfache Struktur hat. Dies erlaubt es sehr allgemeine Aussagen über die Stabilität des synchronen Zustands zu treffen und löst das Problem vollständiger Synchronisation im Limes für lange Verzögerungen. Weiterhin betrachte ich verallgemeinerte Formen von Synchronisation in delay-gekoppelten Systemen. Hierbei interessiere ich mich im Hinblick auf Laser insbesondere für Systeme mit einer Rotationssymmetry. In einem konkretes Lasersystem untersuche ich die Stabilität des synchronisierten Zustandes numerisch durch die Berechnung von transversalen Lyapunov Exponenten. In diesem System tritt Desynchronisation durch "bubbling" auf. Dieser Effekt lässt sich auf einfache Weise durch die Eigenschaften der instabilen Lasermoden erklären.In this thesis I investigate the effect of delay in complex nonlinear systems and its application to laser systems. I concentrate on two main aspects: (i) the noninvasive stabilization of periodic orbits by time-delayed feedback control and here in particular the stabilization of odd-number orbits and (ii) the synchronization of delay coupled systems. In both cases the main question concerns the stability of solutions under the influence of delay. Time-delayed feedback control as proposed by Pyragas has been invented to stabilize periodic orbits. It was the accepted opinion that so-called odd-number orbits, i.e., periodic orbits with an odd number of unstable Floquet multipliers, could not be stabilized with this method. This misbelief was recently refuted. In this thesis I give a detailed discussion of the counter example. Furthermore, I show how to construct novel feedback schemes which successfully stabilize odd-number orbits and which are directly applicable to experiments. These control schemes are then applied to laser systems using normal form analysis and numerical simulations. For delay coupled systems I focus on synchronization phenomena and in particular on the case of chaos synchronization. The most significant result of this part concerns the master stability function for delay coupled networks. Here I show that in the limit of large delay the master stability function has a simple structure which allows to draw very general conclusions about the stability of synchronized solutions. Thus I solve the problem of complete synchronization for systems coupled with a large delay. Furthermore, I consider more generalized types of synchronization and investigate for which delayed coupling schemes synchronized solutions exist. Here I consider individual systems which have a symmetry (mainly rotation symmetry). This is inspired by lasers for which the dynamical equations are invariant under phase shifts. I explicitly derive coupling conditions which allow synchronization and draw the connection to delay-coupled lasers. For a concrete system of two delay coupled lasers with self-feedback I investigate the stability of the synchronized state numerically by calculating transversal Lyapunov exponents. Here, I observe on-off intermittency and bubbling and explain this desynchronization dynamics through properties of the unstable laser modes