Spatio-temporal patterns appear almost everywhere in nature, and their description and understanding still raise important and basic questions. However, if one looks back 20 or 30 years, definite progress has been made in the modeling of insta bilities, analysis of the dynamics in their vicinity, pattern formation and stability, quantitative experimental and numerical analysis of patterns, and so on. Universal behaviors of complex systems close to instabilities have been determined, leading to the wide interdisciplinarity of a field that is now referred to as nonlinear science or science of complexity, and in which initial concepts of dissipative structures or synergetics are deeply rooted. In pioneering domains related to hydrodynamics or chemical instabilities, the interactions between experimentalists and theoreticians, sometimes on a daily basis, have been a key to progress. Everyone in the field praises the role played by the interactions and permanent feedbacks between ex perimental, numerical, and analytical studies in the achievements obtained during these years. Many aspects of convective patterns in normal fluids, binary mixtures or liquid crystals are now understood and described in this framework. The generic pres ence of defects in extended systems is now well established and has induced new developments in the physics of laser with large Fresnel numbers. Last but not least, almost 40 years after his celebrated paper, Turing structures have finally been ob tained in real-life chemical reactors, triggering anew intense activity in the field of reaction-diffusion systems.

The purpose of the NATO Advanced Research Workshop, upon which this book is based, was to bring together experimentalists and theorists from many different fields, ranging from applied mathematics to materials science, but unified by their intrigue with nonlinear phenomena, in search of a deeper understanding of patterns in complex systems. To meet this goal, the participants made the effort to build bridges across canonical disciplinary boundaries by sharing what they thought was significant and relevant in search of the “truly significant simplicity of the basic laws of nature embedded in the amazing complexity of natural phenomena.” Spatio-Temporal Patterns in Nonequilibrium Complex Systems is one of the most exciting and fastest-growing branches of physics that impacts fields as diverse as new technologies and processes, economics and biology. Virtually every structure in our world, including ourselves, can be considered the result of a long sequence of successive symmetry-breaking instabilities due to nonlinear processes under nonequilibrium conditions of a complex system. While a scientific description of the spontaneous appearance of patterns in nature was first made by Johannes Kepler (1611), it has only been during the past twenty years that pattern formation, epitomized by the beautiful snowflakes that Kepler studied, has emerged as a science. Concepts and methods resulting from this dynamic new field will surely influence future developments in many disciplines.Complex systems, as studied in this book, are a good first step toward a description of the variety of phenomena included under the rubric “physics of complex systems.” Even the simplest of those presented here, liquid crystals, is still complex, but provides hints of essential ingredients needed to forge a fundamental understanding of nonequilibrium, nonlinear processes in the large. Fluid dynamics and turbulence, interface motion during solidification, autocatalytic chemical reactions, and pattern formation in biological systems play similar roles in other systems far from equilibrium.

A rich variety of real-life physical problems which are still poorly understood are of a nonlinear nature. Examples include turbulence, granular flows, detonations and flame propagation, fracture dynamics, and a wealth of new biological and chemical phenomena which are being discovered. Particularly interesting among the manifestations of nonlinearity are coherent structures. This book contains reviews and contributions reporting on the state of the art regarding the role of coherent structures and patterns in nonlinear science.

This volume contains a selection of the lectures given at the Fifth International Workshop on Instabilities and Nonequilibrium Structures, held in Santiago, Chile, in December 1993. The following general subjects are covered: instabilities and pattern formation, stochastic effects in nonlinear systems, nonequilibrium statistical mechanics and granular matter. Review articles on transitions between spatio-temporal patterns and nonlinear wave equations are also included. Audience: This book should appeal to physicists and mathematicians working in the areas of nonequilibrium systems, dynamical systems, pattern formation and partial differential equations. Chemists and biologists interested in self-organization and statistical mechanics should also be interested, as well as engineers working in fluid mechanics and materials science.

This book contains a thorough treatment of neural networks, cellular-automata and synergetics, in an attempt to provide three different approaches to nonlinear phenomena in complex systems. These topics are of major interest to physicists active in the fields of statistical mechanics and dynamical systems. They have been developed with a high degree of sophistication and include the refinements necessary to work with the complexity of real systems as well as the more recent research developments in these areas.

The contents of this book correspond to Sessions VII and VIII of the International Workshop on Instabilities and Nonequilibrium Structures which took place in Viña del Mar, Chile, in December 1997 and December 1999, respectively. Part I is devoted to self-contained courses. Three courses are related to new developments in Bose-Einstein condensation: the first one by Robert Graham studies the classical dynamics of excitations of Bose condensates in anisotropic traps, the second by Marc Etienne Brachet refers to the bifurcations arising in attractive Bose-Einstein condensates and superfluid helium and the third course by André Verbeure is a pedagogical introduction to the subject with special emphasis on first principles and rigorous results. Part I is completed by two courses given by Michel Moreau: the first one on diffusion limited reactions of particles with fluctuating activity and the second on the phase boundary dynamics in a one dimensional nonequilibrium lattice gas. Part II includes a selection of invited seminars at both Workshops.

In recent years, scientists have applied the principles of complex systems science to increasingly diverse fields. The results have been nothing short of remarkable: their novel approaches have provided answers to long-standing questions in biology, ecology, physics, engineering, computer science, economics, psychology and sociology. "Unifying Themes in Complex Systems" is a well established series of carefully edited conference proceedings that serve the purpose of documenting and archiving the progress of cross-fertilization in this field. About NECSI: For over 10 years, The New England Complex Systems Institute (NECSI) has been instrumental in the development of complex systems science and its applications. NECSI conducts research, education, knowledge dissemination, and community development around the world for the promotion of the study of complex systems and its application for the betterment of society. NECSI hosts the International Conference on Complex Systems and publishes the NECSI Book Series in conjunction with Springer Publishers.