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Author: Wolfgang Siegert
Publisher: Springer Science & Business Media
ISBN: 3540859632
Category : Mathematics
Languages : en
Pages : 264
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Book Description
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.
Author: Wolfgang Siegert
Publisher: Springer Science & Business Media
ISBN: 3540859632
Category : Mathematics
Languages : en
Pages : 264
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Book Description
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.
Author: A. J. Lichtenberg
Publisher: Springer Science & Business Media
ISBN: 1475742576
Category : Mathematics
Languages : en
Pages : 518
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Book Description
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.
Author: Gottfried Mayer-Kress
Publisher: Springer Science & Business Media
ISBN: 3642710018
Category : Science
Languages : en
Pages : 264
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Book Description
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.
Author: Marcelo Viana
Publisher: Cambridge University Press
ISBN: 1316062694
Category : Mathematics
Languages : en
Pages : 217
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Book Description
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.
Author: Luis Barreira
Publisher: American Mathematical Soc.
ISBN: 0821829211
Category : Mathematics
Languages : en
Pages : 166
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Book Description
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.
Author: Luís Barreira
Publisher: American Mathematical Society
ISBN: 1470473070
Category : Mathematics
Languages : en
Pages : 355
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Book Description
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.
Author: Juan C. Vallejo
Publisher: Springer
ISBN: 3319518933
Category : Science
Languages : en
Pages : 147
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Book Description
This book is primarily concerned with the computational aspects of predictability of dynamical systems – in particular those where observation, modeling and computation are strongly interdependent. Unlike with physical systems under control in laboratories, for instance in celestial mechanics, one is confronted with the observation and modeling of systems without the possibility of altering the key parameters of the objects studied. Therefore, the numerical simulations offer an essential tool for analyzing these systems. With the widespread use of computer simulations to solve complex dynamical systems, the reliability of the numerical calculations is of ever-increasing interest and importance. This reliability is directly related to the regularity and instability properties of the modeled flow. In this interdisciplinary scenario, the underlying physics provide the simulated models, nonlinear dynamics provides their chaoticity and instability properties, and the computer sciences provide the actual numerical implementation. This book introduces and explores precisely this link between the models and their predictability characterization based on concepts derived from the field of nonlinear dynamics, with a focus on the finite-time Lyapunov exponents approach. The method is illustrated using a number of well-known continuous dynamical systems, including the Contopoulos, Hénon-Heiles and Rössler systems. To help students and newcomers quickly learn to apply these techniques, the appendix provides descriptions of the algorithms used throughout the text and details how to implement them in order to solve a given continuous dynamical system.
Author: Paul Bourgine
Publisher: Springer
ISBN: 3319459015
Category : Science
Languages : en
Pages : 407
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Book Description
This book contains the proceedings as well as invited papers for the first annual conference of the UNESCO Unitwin Complex System Digital Campus (CSDC), which is an international initiative gathering 120 Universities on four continents, and structured in ten E-Departments. First Complex Systems Digital Campus World E-Conference 2015 features chapters from the latest research results on theoretical questions of complex systems and their experimental domains. The content contained bridges the gap between the individual and the collective within complex systems science and new integrative sciences on topics such as: genes to organisms to ecosystems, atoms to materials to products, and digital media to the Internet. The conference breaks new ground through a dedicated video-conferencing system – a concept at the heart of the international UNESCO UniTwin, embracing scientists from low-income and distant countries. This book promotes an integrated system of research, education, and training. It also aims at contributing to global development by taking into account its social, economic, and cultural dimensions. First Complex Systems Digital Campus World E-Conference 2015 will appeal to students and researchers working in the fields of complex systems, statistical physics, computational intelligence, and biological physics.
Author: Mark Pollicott
Publisher: Cambridge University Press
ISBN: 9780521435932
Category : Mathematics
Languages : en
Pages : 176
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Book Description
These lecture notes provide a unique introduction to Pesin theory and its applications.
Author: George Contopoulos
Publisher: Springer Science & Business Media
ISBN: 3662049171
Category : Science
Languages : en
Pages : 633
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Book Description
This book is one of the first to provide a general overview of order and chaos in dynamical astronomy. The progress of the theory of chaos has a profound impact on galactic dynamics. It has even invaded celestial mechanics, since chaos was found in the solar system which in the past was considered as a prototype of order. The book provides a unifying approach to these topics from an author who has spent more than 50 years of research in the field. The first part treats order and chaos in general. The other two parts deal with order and chaos in galaxies and with other applications in dynamical astronomy, ranging from celestial mechanics to general relativity and cosmology.
