International Journal of Bifurcation and Chaos

Simple electronic oscillators were at the origin of many studies related to the qualitative theory of dynamical systems. Chua's circuit is now playing an equivalent role for the generation and understanding of complex dynamics.

In honour of my friend Leon Chua on his 60th birthday.

For well defined classes of dynamical systems (chaos, nonintegrable systems in the thermodynamic limit) it is now possible to formulate the basic laws of the probabilistic level. This leads to an extension of classical and quantum theory to include time symmetry breaking. This leads to a unification of the time reversible description of dynamics and the evolutionary view of thermodynamics.

Synergetics is an interdisciplinary field of research. It deals with open systems that are composed of many individual parts that interact with each other and that can form spatial, temporal, or functional structures by self-organization. The research goal of synergetics is three-fold: (1) Are there general principles of self-organization? (2) Are there analogies in the behavior of self-organizing systems? (3) Can new devices be constructed because of the results (1) and (2)? From a mathematical point of view, synergetics deals with nonlinear partial stochastic differential equations and studies their solutions close to those points where the solutions change their behavior qualitatively. As I will show in my article, synergetics in its present form is based on the concepts of stability and instability, control parameters, order parameters, and the slaving principle. The slaving principle allows one to compress the information that is necessary to describe complex systems into a few order parameters. This is possible if systems are close to their instability points. But it appears that the order parameter concept is also applicable to situations away from such instabilities. At the level of the order parameter equations, profound analogies between otherwise quite different systems become visible. This allows one to realize the same process (for instance dealing with information) on quite different material substrates. The order parameter concept and the slaving principle are explained and their extension to discrete noisy maps and to delay equations are mentioned. These results can be applied to pattern formation in fluids, lasers, semiconductors, plasmas, and other fields. A section is devoted to the analysis of spatio-temporal patterns in terms of order parameters and the slaving principle. It is shown how the concepts of synergetics can be utilized to devise a new type of computer for pattern recognition. In connection with preprocessing, it can recognize patterns that are shifted, scaled or rotated in space and that are deformed. It can recognize scenes and also facial expressions as well as movement patterns. The learning procedure is briefly outlined. Because of the analogy principle of synergetics, this computer allows for hardware realizations by means of semiconductors and lasers. Decision-making by humans or machines is interpreted by means of an analogy with pattern recognition. Further sections are devoted to recognition of dynamic processes and learning by machines. In this author's opinion, synergetics will find important applications in medicine, for instance in the analysis of MEG and EEG patterns, and in the development of devices with brain-like functions. Future tasks of synergetics will be the application of the order parameter concept and the slaving principle to the integration of specialized computers or computer algorithms, for instance for the recognition of faces, movement patterns, and so on, into a computer network for scene analysis and decision making. This will also hold for complicated production processes and so on. Generally speaking, the potentialities of synergetics are based on its self-organization principles.

We review the theory of nonlinear systems, especially that of strange attractors, and give its perspectives. Of a special attention are the recent results concerning hyperbolic attractors and features of high-dimensional systems in the Newhouse regions. We present an example of a "wild" strange attractor of the topological dimension three.

The review gives and account of the historical development, the current state and possible future developments of experimental nonlinear physics, with emphasis on acoustics, hydrodynamics and optics. The concepts of nonlinear time-series analysis which are the basis of the analysis of experimental outcomes from nonlinear systems are explained and recent developments pertaining to such different fields as modeling, prediction, nonlinear noise reduction, detecting determinism, synchronization, and spatio-temporal time series are surveyed. An overview is given of experiments on acoustic cavitation, a field rich of nonlinear phenomena such as nonlinear oscillations, chaotic dynamics and structure formation, and one of the first physical systems to exhibit period-doubling and chaos in experiment.

Integrability and chaos are two of the main concepts associated with nonlinear physical systems which have revolutionized our understanding of them. Highly stable exponentially localized solitons are often associated with many of the important integrable nonlinear systems while motions which are sensitively dependent on initial conditions are associated with chaotic systems. Besides dramatically raising our perception of many natural phenomena, these concepts are opening up new vistas of applications and unfolding technologies: Optical soliton based information technology, magnetoelectronics, controlling and synchronization of chaos and secure communications, to name a few. These developments have raised further new interesting questions and potentialities. We present a particular view of some of the challenging problems and payoffs ahead in the next few decades by tracing the early historical events, summarizing the revolutionary era of 1950–1970 when many important new ideas including solitons and chaos were realized and reviewing the current status. Important open problems both at the basic and applied levels are discussed.

Nonlinear science has primarily developed from applications of mathematics to physics. The biological sciences are emerging as the dominant growth points of science and technology, and biological systems are characterized by being information dense, spatially extended, organized in interacting hierarchies, and rich in diversity. These characteristics, linked with an increase in available computing power and accessible memory, may lead to a nonlinear science of complicated interacting systems that will link different types of mathematical objects within a framework of algebraic models of computing systems. Examples, drawn from current work on intracellular, cellular, tissue, organ, and integrative physiology of an individual, are outlined within the theory of synchronous concurrent algorithms. Possible directions in population dynamics and applications to ecosystem management are outlined.

Nonlinear Computation will provide a framework for the computational science and engineering of the future. Topics such as bifurcation and chaos, and methods such as continuation and branch switching, and features such as stability and sensitivity are basic ingredients in the attempt to understand our world. Nonlinear Computation is indispensable when we try to predict and simulate the dynamics of states in the biological systems we consist of, in the ecological and economical systems we live in, and in the technical systems we make use of.

This paper essentially consists of three parts. The first part (Sec. 1) explains what nonlinear computation means. The second part (Secs. 2–5) presents a survey on nonlinear phenomena, and on the methods of nonlinear computation. The final part (Secs. 6 and 7) will speculate on visions and trends for the next decades.

The objectives and some basic methods of numerical bifurcation analysis are described. Several computational examples are used to illustrate the power as well as the limitations of these techniques. Directions in which further algorithmic and software development is desirable are also discussed.

This paper does not pretend to present a comprehensive history of nonlinear dynamics. Its purpose is more modest and limited to some historical aspects of this topic. The first part of this paper deals with the early foundations of nonlinear dynamics (essentially the Poincaré and Lyapunov results). The succeeding sections cover the period 1910–1970 and describes the development and contributions of the theory, elaborated by Birkhoff, the Andronov school, and the Krylov–Bogoliubov school. After 1970, the development of new results in nonlinear dynamics has become "explosive". A part of these results is presented in a summarized form in this paper. The last section suggests some possible trends for future research.