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The Cambridge Nonlinear Science Series is an international series that addresses all aspects of contemporary research in classical and quantum nonlinear dynamics, both deterministic and nondeterministic, at the level of graduate text and monograph. The flavour is intended to be an approximately equal blend of experimental and theoretical works, with the emphasis in the latter on testable results. Specific subject areas suitable for consideration include: Hamiltonian and dissipative chaos; squeezed states and applications of quantum measurement theory; pattern selection, formation and recognition; networks and learning systems; complexity in low- and high-dimensional systems and random noise; cellular automata; fully developed, weak and phase turbulence; reaction-diffusion systems; bifurcation theory and applications; self-structured states leading to chaos; the physics of interfaces, including fractal and multifractal growth; and simulations used in studies of these topics.

  • General Editors: Boris Chirikov, Budker Institute of Nuclear Physics, Novosibirsk, Russia, Predrag Cvitanovic, Niels Bohr Institutet, Copenhagen , Frank Moss, University of Missouri, St Louis , Harry Swinney, University of Texas, Austin

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9 results in Cambridge Nonlinear Science Series

Chaos, Dynamics, and Fractals
  • An Algorithmic Approach to Deterministic Chaos
  • Joseph L. McCauley
  • Published online:
    05 October 2013
    Print publication:
    07 January 1993
  • This book develops deterministic chaos and fractals from the standpoint of iterated maps, but the emphasis makes it very different from all other books in the field. It provides the reader with an introduction to more recent developments, such as weak universality, multifractals, and shadowing, as well as to older subjects like universal critical exponents, devil's staircases and the Farey tree. The author uses a fully discrete method, a 'theoretical computer arithmetic', because finite (but not fixed) precision cannot be avoided in computation or experiment. This leads to a more general formulation in terms of symbolic dynamics and to the idea of weak universality. The connection is made with Turing's ideas of computable numbers and it is explained why the continuum approach leads to predictions that are not necessarily realized in computation or in nature, whereas the discrete approach yields all possible histograms that can be observed or computed.

Synchronization
  • A Universal Concept in Nonlinear Sciences
  • Arkady Pikovsky, Michael Rosenblum, Jürgen Kurths
  • Published online:
    06 July 2010
    Print publication:
    18 October 2001
  • First recognized in 1665 by Christiaan Huygens, synchronization phenomena are abundant in science, nature, engineering and social life. Systems as diverse as clocks, singing crickets, cardiac pacemakers, firing neurons and applauding audiences exhibit a tendency to operate in synchrony. These phenomena are universal and can be understood within a common framework based on modern nonlinear dynamics. The first half of this book describes synchronization without formulae, and is based on qualitative intuitive ideas. The main effects are illustrated with experimental examples and figures, and the historical development is outlined. The remainder of the book presents the main effects of synchronization in a rigorous and systematic manner, describing classical results on synchronization of periodic oscillators, and recent developments in chaotic systems, large ensembles, and oscillatory media. This comprehensive book will be of interest to a broad audience, from graduate students to specialist researchers in physics, applied mathematics, engineering and natural sciences.

Chaos, Scattering and Statistical Mechanics
  • Pierre Gaspard
  • Published online:
    30 January 2010
    Print publication:
    21 May 1998
  • This book describes advances in the application of chaos theory to classical scattering and nonequilibrium statistical mechanics generally, and to transport by deterministic diffusion in particular. The author presents the basic tools of dynamical systems theory, such as dynamical instability, topological analysis, periodic-orbit methods, Liouvillian dynamics, dynamical randomness and large-deviation formalism. These tools are applied to chaotic scattering and to transport in systems near equilibrium and maintained out of equilibrium. Chaotic Scattering is illustrated with disk scatterers and with examples of unimolecular chemical reactions and then generalized to transport in spatially extended systems. This book will be bought by researchers interested in chaos, dynamical systems, chaotic scattering, and statistical mechanics in theoretical, computational and mathematical physics and also in theoretical chemistry.

Dynamical Systems Approach to Turbulence
  • Tomas Bohr, Mogens H. Jensen, Giovanni Paladin, Angelo Vulpiani
  • Published online:
    23 December 2009
    Print publication:
    13 August 1998
  • This book, first published in 1998, treats turbulence from the point of view of dynamical systems. The exposition centres around a number of important simplified models for turbulent behaviour in systems ranging from fluid motion (classical turbulence) to chemical reactions and interfaces in disordered systems.The modern theory of fractals and multifractals now plays a major role in turbulence research, and turbulent states are being studied as important dynamical states of matter occurring also in systems outside the realm of hydrodynamics, i.e. chemical reactions or front propagation. The presentation relies heavily on simplified models of turbulent behaviour, notably shell models, coupled map lattices, amplitude equations and interface models, and the focus is primarily on fundamental concepts such as the differences between large and small systems, the nature of correlations and the origin of fractals and of scaling behaviour. This book will be of interest to graduate students and researchers interested in turbulence, from physics and applied mathematics backgrounds.

Weak Chaos and Quasi-Regular Patterns
  • Georgin Moiseevich Zaslavskiî, R. Z. Sagdeev, D. A. Usikov, A. A. Chernikov
  • Translated by A. R. Sagdeeva
  • Published online:
    29 October 2009
    Print publication:
    11 April 1991
  • This book, the first in the Cambridge Nonlinear Science Series, presents the fundamentals of chaos theory in conservative systems, providing a systematic study of the theory of transitional states of physical systems which lie between deterministic and chaotic behaviour. The authors' treatment of transitions to chaos, the theory of stochastic layers and webs, and the numerous applications of this theory, particularly to pattern symmetry, will make the book of importance to scientists from many disciplines.The authors have been meticulous in providing a detailed presentation of the material, enabling the reader to learn the necessary computational methods and to apply them in other problems. The inclusion of a significant amount of computer graphics is also an important aid to understanding. The final section of the book contains a fascinating collection of patterns in art and living nature. The book will be of interest to graduate students and researchers in physics and mathematics who are interested in problems of chaos, irreversibility, statistical mechanics and theories of spatial patterns and symmetries. The perhaps unconventional links between chaos theory and other topics make the book particularly interesting.

Complexity
  • Hierarchical Structures and Scaling in Physics
  • Remo Badii, Antonio Politi
  • Published online:
    20 October 2009
    Print publication:
    03 April 1997
  • This is a comprehensive discussion of complexity as it arises in physical, chemical, and biological systems, as well as in mathematical models of nature. Common features of these apparently unrelated fields are emphasised and incorporated into a uniform mathematical description, with the support of a large number of detailed examples and illustrations. The quantitative study of complexity is a rapidly developing subject with special impact in the fields of physics, mathematics, information science, and biology. Because of the variety of the approaches, no comprehensive discussion has previously been attempted. This book will be of interest to graduate students and researchers in physics (nonlinear dynamics, fluid dynamics, solid-state, cellular automata, stochastic processes, statistical mechanics and thermodynamics), mathematics (dynamical systems, ergodic and probability theory), information and computer science (coding, information theory and algorithmic complexity), electrical engineering and theoretical biology.

Lattice Gas Hydrodynamics
  • J.-P. Rivet, J. P. Boon
  • Published online:
    13 October 2009
    Print publication:
    04 January 2001
  • Lattice Gas Hydrodynamics describes the approach to fluid dynamics using a micro-world constructed as an automaton universe, where the microscopic dynamics is based not on a description of interacting particles, but on the laws of symmetry and invariance of macroscopic physics. We imagine point-like particles residing on a regular lattice, where they move from node to node and undergo collisions when their trajectories meet. If the collisions occur according to some simple logical rules, and if the lattice has the proper symmetry, then the automaton shows global behavior very similar to that of real fluids. This book carries two important messages. First, it shows how an automaton universe with simple microscopic dynamics - the lattice gas - can exhibit macroscopic behavior in accordance with the phenomenological laws of classical physics. Second, it demonstrates that lattice gases have spontaneous microscopic fluctuations which capture the essentials of actual fluctuations in real fluids.

Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors
  • Eckehard Schöll
  • Published online:
    17 September 2009
    Print publication:
    22 February 2001
  • Nonlinear transport phenomena are an increasingly important aspect of modern semiconductor research. Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors deals with complex nonlinear dynamics, pattern formation, and chaotic behaviour in such systems. In doing so it bridges the gap between two well-established fields: the theory of dynamic systems, and nonlinear charge transport in semiconductors. This unified approach is used to consider important electronic transport instabilities. The initial chapters lay a general framework for the theoretical description of nonlinear self-organized spatio-temporal patterns, like current filaments, field domains, fronts, and analysis of their stability. Later chapters consider important model systems in detail: impact ionization induced impurity breakdown, Hall instabilities, superlattices, and low-dimensional structures. State-of-the-art results include chaos control, spatio-temporal chaos, multistability, pattern selection, activator-inhibitor kinetics, and global coupling, linking fundamental issues to electronic device applications. This book will be of great value to semiconductor physicists and nonlinear scientists alike.

Thermodynamics of Chaotic Systems
  • An Introduction
  • Christian Beck, Friedrich Schögl
  • Published online:
    14 September 2009
    Print publication:
    01 July 1993
  • This book deals with the various thermodynamic concepts used for the analysis of nonlinear dynamical systems. The most important invariants used to characterise chaotic systems are introduced in a way that stresses the interconnections with thermodynamics and statistical mechanics. Among the subjects treated are: probabilistic aspects of chaotic dynamics; the symbolic dynamics technique; information measures, the maximum entropy principle; general thermodynamic relations; spin systems; fractals and multifractals; expansion rate and information loss; the topological pressure; transfer operator methods; and repellers and escape. The more advanced chapters deal with the thermodynamic formalism for expanding maps, the thermodynamic analysis of chaotic systems with several intensive parameters and phase transitions in nonlinear dynamics.