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Synchronization: A Universal Concept in Nonlinear Sciences
01 Jan 2001-
TL;DR: This work discusseschronization of complex dynamics by external forces, which involves synchronization of self-sustained oscillators and their phase, and its applications in oscillatory media and complex systems.
Abstract: Preface 1. Introduction Part I. Synchronization Without Formulae: 2. Basic notions: the self-sustained oscillator and its phase 3. Synchronization of a periodic oscillator by external force 4. Synchronization of two and many oscillators 5. Synchronization of chaotic systems 6. Detecting synchronization in experiments Part II. Phase Locking and Frequency Entrainment: 7. Synchronization of periodic oscillators by periodic external action 8. Mutual synchronization of two interacting periodic oscillators 9. Synchronization in the presence of noise 10. Phase synchronization of chaotic systems 11. Synchronization in oscillatory media 12. Populations of globally coupled oscillators Part III. Synchronization of Chaotic Systems: 13. Complete synchronization I: basic concepts 14. Complete synchronization II: generalizations and complex systems 15. Synchronization of complex dynamics by external forces Appendix 1. Discovery of synchronization by Christiaan Huygens Appendix 2. Instantaneous phase and frequency of a signal References Index.
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Abstract: The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.
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Cites background from "Synchronization: A Universal Concep..."
...Synchronization [272] is an emergent phenomenon occurring in systems of interacting units and is ubiquitous in nature, society and technology....
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2,993 citations
Cites background or methods or result from "Synchronization: A Universal Concep..."
...This is in agreement with the criterion for PS via Lyapunov exponents i [133]: 4 becomes negative at ≈ 0....
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...Increasing the coupling strength, PS can be obtained by the transition of one of the zero Lyapunov exponents to negative values, indicating the establishment of a relationship between the phases [133]....
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..., the analytical signal approach [133] and compare them with sj i = | xi − sj yi |....
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...33; the mean frequencies 1 and 2 are calculated as proposed in [133])....
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...Then the phase can be identified with the angle of rotation [133,148]...
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TL;DR: In this paper, a review of the Kuramoto model of coupled phase oscillators is presented, with a rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years.
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2,864 citations
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