Synchrony and bifurcations in coupled dynamical systems and effects of time delay

Dynamik auf Netzwerken ist ein mathematisches Feld, das in den letzten Jahrzehnten schnell gewachsen ist und Anwendungen in zahlreichen Disziplinen wie z.B. Physik, Biologie und Soziologie findet. Die Funktion vieler Netzwerke hängt von der Fähigkeit ab, die Elemente des Netzwerkes zu synchronisieren. Mit anderen Worten, die Existenz und die transversale Stabilität der synchronen Mannigfaltigkeit sind zentrale Eigenschaften. Erst seit einigen Jahren wird versucht, den verwickelten Zusammenhang zwischen der Kopplungsstruktur und den Stabilitätseigenschaften synchroner Zustände zu verstehen. Genau das ist das zentrale Thema dieser Arbeit. Zunächst präsentiere ich erste Ergebnisse zur Klassifizierung der Kanten eines gerichteten Netzwerks bezüglich ihrer Bedeutung für die Stabilität des synchronen Zustands. Folgend untersuche ich ein komplexes Verzweigungsszenario in einem gerichteten Ring von Stuart-Landau Oszillatoren und zeige, dass das Szenario persistent ist, wenn dem Netzwerk eine schwach gewichtete Kante hinzugefügt wird. Daraufhin untersuche ich synchrone Zustände in Ringen von Phasenoszillatoren die mit Zeitverzögerung gekoppelt sind. Ich bespreche die Koexistenz synchroner Lösungen und analysiere deren Stabilität und Verzweigungen. Weiter zeige ich, dass eine Zeitverschiebung genutzt werden kann, um Muster im Ring zu speichern und wiederzuerkennen. Diese Zeitverschiebung untersuche ich daraufhin für beliebige Kopplungsstrukturen. Ich zeige, dass invariante Mannigfaltigkeiten des Flusses sowie ihre Stabilität unter der Zeitverschiebung erhalten bleiben. Darüber hinaus bestimme ich die minimale Anzahl von Zeitverzögerungen, die gebraucht werden, um das System äquivalent zu beschreiben. Schließlich untersuche ich das auffällige Phänomen eines nichtstetigen Übergangs zu Synchronizität in Klassen großer Zufallsnetzwerke indem ich einen kürzlich eingeführten Zugang zur Beschreibung großer Zufallsnetzwerke auf den Fall zeitverzögerter Kopplungen verallgemeinere.Since a couple of decades, dynamics on networks is a rapidly growing branch of mathematics with applications in various disciplines such as physics, biology or sociology. The functioning of many networks heavily relies on the ability to synchronize the network’s nodes. More precisely, the existence and the transverse stability of the synchronous manifold are essential properties. It was only in the last few years that people tried to understand the entangled relation between the coupling structure of a network, given by a (di-)graph, and the stability properties of synchronous states. This is the central theme of this dissertation. I first present results towards a classification of the links in a directed, diffusive network according to their impact on the stability of synchronization. Then I investigate a complex bifurcation scenario observed in a directed ring of Stuart-Landau oscillators. I show that under the addition of a single weak link, this scenario is persistent. Subsequently, I investigate synchronous patterns in a directed ring of phase oscillators coupled with time delay. I discuss the coexistence of multiple of synchronous solutions and investigate their stability and bifurcations. I apply these results by showing that a certain time-shift transformation can be used in order to employ the ring as a pattern recognition device. Next, I investigate the same time-shift transformation for arbitrary coupling structures in a very general setting. I show that invariant manifolds of the flow together with their stability properties are conserved under the time-shift transformation. Furthermore, I determine the minimal number of delays needed to equivalently describe the system’s dynamics. Finally, I investigate a peculiar phenomenon of non-continuous transition to synchrony observed in certain classes of large random networks, generalizing a recently introduced approach for the description of large random networks to the case of delayed couplings

Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed

New Trends in Statistical Physics of Complex Systems

A topical research activity in statistical physics concerns the study of complex and disordered systems. Generally, these systems are characterized by an elevated level of interconnection and interaction between the parts so that they give rise to a rich structure in the phase space that self-organizes under the control of internal non-linear dynamics. These emergent collective dynamics confer new behaviours to the whole system that are no longer the direct consequence of the properties of the single parts, but rather characterize the whole system as a new entity with its own features, giving rise to the birth of new phenomenologies. As is highlighted in this collection of papers, the methodologies of statistical physics have become very promising in understanding these new phenomena. This volume groups together 12 research works showing the use of typical tools developed within the framework of statistical mechanics, in non-linear kinetic and information geometry, to investigate emerging features in complex physical and physical-like systems. A topical research activity in statistical physics concerns the study of complex and disordered systems. Generally, these systems are characterized by an elevated level of interconnection and interaction between the parts so that they give rise to a rich structure in the phase space that self-organizes under the control of internal non-linear dynamics. These emergent collective dynamics confer new behaviours to the whole system that are no longer the direct consequence of the properties of the single parts, but rather characterize the whole system as a new entity with its own features, giving rise to the birth of new phenomenologies. As is highlighted in this collection of papers, the methodologies of statistical physics have become very promising in understanding these new phenomena. This volume groups together 12 research works showing the use of typical tools developed within the framework of statistical mechanics, in non-linear kinetic and information geometry, to investigate emerging features in complex physical and physical-like systems

Theoretical Concepts of Quantum Mechanics

Quantum theory as a scientific revolution profoundly influenced human thought about the universe and governed forces of nature. Perhaps the historical development of quantum mechanics mimics the history of human scientific struggles from their beginning. This book, which brought together an international community of invited authors, represents a rich account of foundation, scientific history of quantum mechanics, relativistic quantum mechanics and field theory, and different methods to solve the Schrodinger equation. We wish for this collected volume to become an important reference for students and researchers