Synchronization: From Simple to Complex (Springer Series in Synergetics)

**Synchronization: From Simple to Complex** is devoted to the fundamental phenomenon in physics – synchronization that occurs in coupled non-linear dissipative oscillators. Examples of such systems range from mechanical clocks to population dynamics, from human heart to neural networks. The authors study this phenomenon as applied to oscillations of different nature such as those with periodic, chaotic, noisy and noise-induced nature, reveal the general mechanisms behind synchronization, and bring to light other important effects that accompany synchronization such as phase multistability, dephasing and multimode interaction. The main purpose of this book is to demonstrate that the complexity of synchronous patterns of real oscillating system can be described in the framework of the general approach. The homology of manifolds enjoys a remarkable symmetry: Poincaré duality. If the manifold is triangulated, then this duality can be established by associating to a s- plex its dual block in the barycentric subdivision. In a manifold, the dual block is a cell, so the chain complex based on the dual blocks computes the homology of the manifold. Poincaré duality then serves as a cornerstone of manifold classi cation theory. One reason is that it enables the de nition of a fundamental bordism inva- ant, the signature. Classifying manifolds via the surgery program relies on modifying a manifold by executing geometric surgeries. The trace of the surgery is a bordism between the original manifold and the result of surgery. Since the signature is a b- dism invariant, it does not change under surgery and is thus a basic obstruction to performing surgery. Inspired by Hirzebruch's signature theorem, a method of Thom constructs characteristic homology classes using the bordism invariance of the s- nature. These classes are not in general homotopy invariants and consequently are ne enough to distinguish manifolds within the same homotopy type. Singular spaces do not enjoy Poincaré duality in ordinary homology. After all, the dual blocks are not cells anymore, but cones on spaces that may not be spheres. This book discusses when, and how, the invariants for manifolds described above can be established for singular spaces. The Central Theme Of This Book Is The Restoration Of Poincaré Duality On Stratified Singular Spaces By Using Verdier-self-dual Sheaves Such As The Prototypical Intersection Chain Sheaf On A Complex Variety. After Carefully Introducing Sheaf Theory, Derived Categories, Verdier Duality, Stratification Theories, Intersection Homology, T-structures And Perverse Sheaves, The Ultimate Objective Is To Explain The Construction As Well As Algebraic And Geometric Properties Of Invariants Such As The Signature And Characteristic Classes Effectuated By Self-dual Sheaves. Highlights Never Before Presented In Book Form Include Complete And Very Detailed Proofs Of Decomposition Theorems For Self-dual Sheaves, Explanation Of Methods For Computing Twisted Characteristic Classes And An Introduction To The Author's Theory Of Non-witt Spaces And Lagrangian Structures. Elementary Sheaf Theory -- Homological Algebra -- Verdier Duality -- Intersection Homology -- Characteristic Classes And Smooth Manifolds -- Invariants Of Witt Spaces -- T-structures -- Methods Of Computation -- Invariants Of Non-witt Spaces -- L2 Cohomology. M. Banagl. Includes Bibliographical References (p. [251]-253) And Index. Front Matter....Pages i-xvii Introduction....Pages 1-3 Front Matter....Pages 5-5 General Remarks....Pages 9-19 1 : 1 Forced Synchronization of Periodic Oscillations....Pages 21-73 1 : 1 Mutual Synchronization of Periodic Oscillations....Pages 75-104 Homoclinic Mechanism of Synchronization of Periodic Oscillations....Pages 105-120 n : m Synchronization of Periodic Oscillations....Pages 121-148 1 : 1 Forced Synchronization of Periodic Oscillations in the Presence of Noise....Pages 149-190 Chaos Synchronization....Pages 191-238 Synchronization of Noise-Induced Oscillations....Pages 239-258 Conclusions to Part I....Pages 259-260 Front Matter....Pages 261-261 Synchronization of Anisochronous Oscillators....Pages 265-315 Phase Multistability....Pages 317-352 Synchronization in Systems with Complex Multimode Dynamics....Pages 353-376 Synchronization of Systems with Resource Mediated Coupling....Pages 377-409 Conclusions to Part II....Pages 411-412 Back Matter....Pages 413-425 "The central theme of this book is the restoration of Poincare duality on stratified singular spaces by using Verdier-self-dual sheaves such as the prototypical intersection chain sheaf on a complex variety." "After carefully introducing sheaf theory, derived categories, Verdier duality, stratification theories, intersection homology, t-structures and perverse sheaves, the ultimate objective is to explain the construction as well as algebraic and geometric properties of invariants such as the signature and characteristic classes effectuated by self-dual sheaves." "Highlights never before presented in book form include complete and very detailed proofs of decomposition theorems for self-dual sheaves, explanation of methods for computing twisted characteristic classes and an introduction to the author's theory of non-Witt spaces and Lagrangian structures."--Jacket Part I: General Mechanisms Of Synchronization -- General Remarks -- Forced Synchronization Of Periodic Oscillations -- Mutual Synchronization Of Periodic Oscillations -- Homoclinic Mechanism Of Synchronization Of Periodic Oscillations -- N:m Synchronization Of Periodic Oscillations -- Forced Synchronization Of Periodic Oscillations In The Presence Of Noise -- Chaos Synchronization -- Synchronization Of Noise-induced Oscillations -- Conclusions To Part I -- Part Ii: Case Studies In Synchronization -- Synchronization Of Anisochronous Oscillators -- Phase Multistability -- Synchronization In Systems With Complex Multimode Dynamics -- Synchronization Of Systems With Resource Mediated Coupling -- Conclusions To Part Ii. Alexander Balanov ... [et Al.]. Includes Bibliographical References (p. [413]-421) And Index. This Fascinating Work Is Devoted To The Fundamental Phenomenon In Physics – Synchronization That Occurs In Coupled Non-linear Dissipative Oscillators. Examples Of Such Systems Range From Mechanical Clocks To Population Dynamics, From The Human Heart To Neural Networks. The Main Purpose Of This Book Is To Demonstrate That The Complexity Of Synchronous Patterns Of Real Oscillating Systems Can Be Described In The Framework Of The General Approach, And The Authors Study This Phenomenon As Applied To Oscillations Of Different Types, Such As Those With Periodic, Chaotic, Noisy And Noise-induced Nature.