Alexander S. Mikhailov

Synchronization processes bring about dynamical order and lead to spontaneous development of structural organization in complex systems of various origins, from chemical oscillators and biological cells to human societies and the brain. This book provides a review and a detailed theoretical analysis of synchronization phenomena in complex systems with different architectures, composed of elements with periodic or chaotic individual dynamics. Special attention is paid to statistical concepts, such as nonequilibrium phase transitions, order parameters and dynamical glasses.

This book provides an outline of theoretical concepts and their experimental verification in studies of self-organization phenomena in chemical systems, as they emerged in the mid-20th century and have evolved since. Presenting essays on selected topics, it was prepared by authors who have made profound contributions to the field. Traditionally, physical chemistry has been concerned with interactions between atoms and molecules that produce a variety of equilibrium structures - or the 'dead' order - in a stationary state. But biological cells exhibit a different 'living' kind of order, prompting E. Schrödinger to pose his famous question “What is life?” in 1943. Through an unprecedented theoretical and experimental development, it was later revealed that biological self-organization phenomena are in complete agreement with the laws of physics, once they are applied to a special class of thermodynamically open systems and non-equilibrium states. This knowledge has in turn led tothe design and synthesis of simple inorganic systems capable of self-organization effects. These artificial 'living organisms' are able to operate on macroscopic to microscopic scales, even down to single-molecule machines. In the future, such research could provide a basis for a technological breakthrough, comparable in its impact with the invention of lasers and semiconductors. Its results can be used to control natural chemical processes, and to design artificial complex chemical processes with various functionalities. The book offers an extensive discussion of the history of research on complex chemical systems and its future prospects.

This book provides an outline of theoretical concepts and their experimental verification in studies of self-organization phenomena in chemical systems, as they emerged in the mid-20th century and have evolved since. Presenting essays on selected topics, it was prepared by authors who have made profound contributions to the field. Traditionally, physical chemistry has been concerned with interactions between atoms and molecules that produce a variety of equilibrium structures - or the 'dead' order - in a stationary state. But biological cells exhibit a different 'living' kind of order, prompting E. Schrödinger to pose his famous question “What is life?” in 1943. Through an unprecedented theoretical and experimental development, it was later revealed that biological self-organization phenomena are in complete agreement with the laws of physics, once they are applied to a special class of thermodynamically open systems and non-equilibrium states. This knowledge has in turn led tothe design and synthesis of simple inorganic systems capable of self-organization effects. These artificial 'living organisms' are able to operate on macroscopic to microscopic scales, even down to single-molecule machines. In the future, such research could provide a basis for a technological breakthrough, comparable in its impact with the invention of lasers and semiconductors. Its results can be used to control natural chemical processes, and to design artificial complex chemical processes with various functionalities. The book offers an extensive discussion of the history of research on complex chemical systems and its future prospects.

This textbook is based on a lecture course in synergetics given at the University of Moscow. In this second of two volumes, we discuss the emergence and properties of complex chaotic patterns in distributed active systems. Such patterns can be produced autonomously by a system, or can result from selective amplification of fluctuations caused by external weak noise. Although the material in this book is often described by refined mathematical theories, we have tried to avoid a formal mathematical style. Instead of rigorous proofs, the reader will usually be offered only "demonstrations" (the term used by Prof. V. I. Arnold) to encourage intuitive understanding of a problem and to explain why a particular statement seems plausible. We also refrained from detailing concrete applications in physics or in other scientific fields, so that the book can be used by students of different disciplines. While preparing the lecture course and producing this book, we had intensive discussions with and asked the advice of Prof. V. I. Arnold, Prof. S. Grossmann, Prof. H. Haken, Prof. Yu. L. Klimontovich, Prof. R. L. Stratonovich and Prof. Ya.

The second edition of this volume has been extensively revised. A different version of Chap. 7, reflecting recent significant progress in understanding of spatiotempo­ ral chaos, is now provided. Much new material has been included in the sections dealing with intermittency in birth-death models and noise-induced phase transi­ tions. A new section on control of chaotic behavior has been added to Chap. 6. The subtitle of the volume has been changed to better reflect its contents. We acknowledge stimulating discussions with H. Haken and E. Scholl and are grateful to our colleagues M. Bar, D. Battogtokh, M. Eiswirth, M. Hildebrand, K. Krischer, and V. Tereshko for their comments and assistance. We thank M. Lubke for her help in producing new figures for this volume. Berlin and Moscow A. s. Mikhailov April 1996 A. Yu. Loskutov Preface to the First Edition This textbook is based on a lecture course in synergetics given at the University of Moscow. In this second of two volumes, we discuss the emergence and properties of complex chaotic patterns in distributed active systems. Such patterns can be produced autonomously by a system, or can result from selective amplification of fluctuations caused by external weak noise.

This book gives an introduction to the mathematical theory of cooperative behavior in active systems of various origins, both natural and artificial. It is based on a lecture course in synergetics which I held for almost ten years at the University of Moscow. The first volume deals mainly with the problems of pattern fonnation and the properties of self-organized regular patterns in distributed active systems. It also contains a discussion of distributed analog information processing which is based on the cooperative dynamics of active systems. The second volume is devoted to the stochastic aspects of self-organization and the properties of self-established chaos. I have tried to avoid delving into particular applications. The primary intention is to present general mathematical models that describe the principal kinds of coopera­ tive behavior in distributed active systems. Simple examples, ranging from chemical physics to economics, serve only as illustrations of the typical context in which a particular model can apply. The manner of exposition is more in the tradition of theoretical physics than of in mathematics: Elaborate fonnal proofs and rigorous estimates are often replaced the text by arguments based on an intuitive understanding of the relevant models. Because of the interdisciplinary nature of this book, its readers might well come from very diverse fields of endeavor. It was therefore desirable to minimize the re­ quired preliminary knowledge. Generally, a standard university course in differential calculus and linear algebra is sufficient.

This book, written by two well-known scientists, represents an excellent ad­ dition to the Springer Series in Synergetics in several ways. It shows how by rather simple models we can gain remarkable insights into the behavior of complex systems. At the same time it demonstrates the progress made in this interdisciplinary field. While in the early days of Synergetics, the self­ organized coherent action of atoms in the laser - a physical device - was in the foreground of interest (cf. my book Synergetics: An Introduction (Springer, Berlin, Heidelberg, New York 1977)), the coherent action of nerve cells got into the focus of research, as is witnessed by the book by P. Tass in this series (P. Tass, Phase Resetting in Medicine and Biology (Springer, Berlin, Heidel­ berg, New York 1999)). In these books the elements were disturbed by noise. Now, in the present book by Mikhailov and Calenbuhr, the self-organized coherent action of otherwise chaotic elements is studied and important as well as surprising results by Kaneko, Mikhailov and others are presented. Let me mention just another highly interesting problem treated in this book: the coherent interaction of tens of thousands of reactions going on in biological cells. But other phenomena, such as the formation of swarms of fish or the collective behavior of ants, are also modelled. These are just a few examples of the many fascinating subjects dealt with in this book that relate to many disciplines under unifying aspects.

This book, written by two well-known scientists, represents an excellent ad­ dition to the Springer Series in Synergetics in several ways. It shows how by rather simple models we can gain remarkable insights into the behavior of complex systems. At the same time it demonstrates the progress made in this interdisciplinary field. While in the early days of Synergetics, the self­ organized coherent action of atoms in the laser - a physical device - was in the foreground of interest (cf. my book Synergetics: An Introduction (Springer, Berlin, Heidelberg, New York 1977)), the coherent action of nerve cells got into the focus of research, as is witnessed by the book by P. Tass in this series (P. Tass, Phase Resetting in Medicine and Biology (Springer, Berlin, Heidel­ berg, New York 1999)). In these books the elements were disturbed by noise. Now, in the present book by Mikhailov and Calenbuhr, the self-organized coherent action of otherwise chaotic elements is studied and important as well as surprising results by Kaneko, Mikhailov and others are presented. Let me mention just another highly interesting problem treated in this book: the coherent interaction of tens of thousands of reactions going on in biological cells. But other phenomena, such as the formation of swarms of fish or the collective behavior of ants, are also modelled. These are just a few examples of the many fascinating subjects dealt with in this book that relate to many disciplines under unifying aspects.