Establishing a new concept of local Lyapunov exponents the author brings together two separate theories, namely Lyapunov exponents and the theory of large deviations. Specifically, a linear differential system is considered which is controlled by a stochastic process that during a suitable noise-intensity-dependent time is trapped near one of its so-called metastable states. The local Lyapunov exponent is then introduced as the exponential growth rate of the linear system on this time scale. Unlike classical Lyapunov exponents, which involve a limit as time increases to infinity in a fixed system, here the system itself changes as the noise intensity converges, too.

These proceedings contain the papers contributed to the International Work shop on "Dimensions and Entropies in Chaotic Systems" at the Pecos River Conference Center on the Pecos River Ranch in Spetember 1985. The work shop was held by the Center for Nonlinear Studies of the Los Alamos National Laboratory. At the Center for Nonlinear Studies the investigation of chaotic dynamics and especially the quantification of complex behavior has a long tradition. In spite of some remarkable successes, there are fundamental, as well as nu merical, problems involved in the practical realization of these algorithms. This has led to a series of publications in which modifications and improve ments of the original methods have been proposed. At present there exists a growing number of competing dimension algorithms but no comprehensive review explaining how they are related. Further, in actual experimental ap plications, rather than a precise algorithm, one finds frequent use of "rules of thumb" together with error estimates which, in many cases, appear to be far too optimistic. Also it seems that questions like "What is the maximal dimension of an attractor that one can measure with a given number of data points and a given experimental resolution?" have still not been answered in a satisfactory manner for general cases.

This book treats stochastic motion in nonlinear oscillator systems. It describes a rapidly growing field of nonlinear mechanics with applications to a number of areas in science and engineering, including astronomy, plasma physics, statistical mechanics and hydrodynamics. The main em phasis is on intrinsic stochasticity in Hamiltonian systems, where the stochastic motion is generated by the dynamics itself and not by external noise. However, the effects of noise in modifying the intrinsic motion are also considered. A thorough introduction to chaotic motion in dissipative systems is given in the final chapter. Although the roots of the field are old, dating back to the last century when Poincare and others attempted to formulate a theory for nonlinear perturbations of planetary orbits, it was new mathematical results obtained in the 1960's, together with computational results obtained using high speed computers, that facilitated our new treatment of the subject. Since the new methods partly originated in mathematical advances, there have been two or three mathematical monographs exposing these developments. However, these monographs employ methods and language that are not readily accessible to scientists and engineers, and also do not give explicit tech niques for making practical calculations. In our treatment of the material, we emphasize physical insight rather than mathematical rigor. We present practical methods for describing the motion, for determining the transition from regular to stochastic behavior, and for characterizing the stochasticity. We rely heavily on numerical computations to illustrate the methods and to validate them.

The theory of Lyapunov exponents originated over a century ago in the study of the stability of solutions of differential equations. Written by one of the subject's leading authorities, this book is both an account of the classical theory, from a modern view, and an introduction to the significant developments relating the subject to dynamical systems, ergodic theory, mathematical physics and probability. It is based on the author's own graduate course and is reasonably self-contained with an extensive set of exercises provided at the end of each chapter. This book makes a welcome addition to the literature, serving as a graduate text and a valuable reference for researchers in the field.

A systematic introduction to the core of smooth ergodic theory. An expanded version of an earlier work by the same authors, it describes the general (abstract) theory of Lyapunov exponents and the theory's applications to the stability theory of differential equations, the stable manifold theory, absolute continuity of stable manifolds, and the ergodic theory of dynamical systems with nonzero Lyapunov exponents (including geodesic flows). It could be used as a primary text for a course on nonuniform hyperbolic theory or as supplemental reading for a course on dynamical systems. Assumes a basic knowledge of real analysis, measure theory, differential equations, and topology. c. Book News Inc.

This book is the first comprehensive introduction to smooth ergodic theory. It consists of two parts: the first introduces the core of the theory and the second discusses more advanced topics. In particular, the book describes the general theory of Lyapunov exponents and its applications to the stability theory of differential equations, the concept of nonuniform hyperbolicity, stable manifold theory (with emphasis on absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. A detailed description of all the basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature, is also presented. There are more than 80 exercises. The book is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wishes to get a working knowledge of smooth ergodic theory and to learn how to use its tools. It can also be used as a source for special topics courses on nonuniform hyperbolicity. The only prerequisite for using this book is a basic knowledge of real analysis, measure theory, differential equations, and topology, although the necessary background definitions and results are provided. In this second edition, the authors improved the exposition and added more exercises to make the book even more student-oriented. They also added new material to bring the book more in line with the current research in dynamical systems.

Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept. Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances in applications to complex systems. Practical algorithms are thoroughly reviewed and their performance is discussed, while a broad set of examples illustrate the wide range of potential applications. The description of various numerical and analytical techniques for the computation of Lyapunov exponents offers an extensive array of tools for the characterization of phenomena such as synchronization, weak and global chaos in low and high-dimensional set-ups, and localization. This text equips readers with all the investigative expertise needed to fully explore the dynamical properties of complex systems, making it ideal for both graduate students and experienced researchers.