Dissipative Structures and Chaos

ISBN-13: 9783642803789 / Angielski / Miękka / 2011 / 299 str.

Hazime Mori; Yoshiki Kuramoto; G. C. Paquette

Opis

This book consists of two parts, the first dealing with dissipative structures and the second with the structure and physics of chaos. The first part was written by Y. Kuramoto and the second part by H. Mori. Throughout the book, emphasis is laid on fundamental concepts and methods rather than applications, which are too numerous to be treated here. Typical physical examples, however, including nonlinear forced oscilla tors, chemical reactions with diffusion, and Benard convection in horizontal fluid layers, are discussed explicitly. Our consideration of dissipative structures is based on a phenomenolog ical reduction theory in which universal aspects of the phenomena under consideration are emphasized, while the theory of chaos is developed to treat transport phenomena, such as the mixing and diffusion of chaotic orbits, from the viewpoint of the geometrical phase space structure of chaos. The title of the original, Japanese version of the book is Sanitsu Kozo to Kaosu (Dissipative Structures and Chaos). It is part of the Iwanami Koza Gendai no Butsurigaku (Iwanami Series on Modern Physics). The first Japanese edition was published in March 1994 and the second in August 1997. We are pleased that this book has been translated into English and that it can now have an audience outside of Japan. We would like to express our gratitude to Glenn Paquette for his English translation, which has made this book more understandable than the original in many respects.

Recenzje

"...extremely well written ...and has sufficient detail and clarity to enable readers (presumably post-graduates) to learn the important techniques presented... the best reference that I know that describes the physics (and in particular the statistical physics) of dissipative structures and chaos...an absolute must" Australian & New Zealand Physicist

Spis treści

I. Dissipative Structures.- 1. A Representative Example of Dissipative Structure.- 1.1 Bénard Convection.- 1.2 The Belousov-Zhabotinskii Reaction.- 2. Amplitude Equations and Their Applications.- 2.1 The Newell-Whitehead Equation and the Stability of Periodic Solutions.- 2.2 Anisotropic Fluids and the Ginzburg-Pitaevskii Equation.- 2.3 Topological Defects and Their Motion.- 2.4 The Amplitude Equation of an Oscillating Field.- 2.5 The Properties of the Complex Ginzburg-Landau Equation.- 2.5.1 Propagating Planar Wave Solutions and Their Stability.- 2.5.2 Rotating Spiral Waves.- 2.5.3 Hole Solutions and Disordered Patterns.- 3. Reaction-Diffusion Systems and Interface Dynamics.- 3.1 Interfaces in Single-Component Bistable Systems.- 3.2 Solitary Wave Pulses and Periodic Wave Pulse Trains in Excitable Systems.- 3.3 Spiral Waves in Excitable Systems.- 3.4 Multiple Spiral Waves and the Turing Pattern.- 3.4.1 Compound Spiral Rotation.- 3.4.2 The Turing Pattern.- 3.5 The Instability of Interfaces and Formation of Structure.- 4. Phase Dynamics.- 4.1 Weak Turbulence of Periodic Structures and the Phase Equation.- 4.2 Phase Waves and Phase Turbulence of Oscillating Fields.- 4.2.1 The Phase Equation of an Oscillating Field and Its Applications.- 4.2.2 Phase Waves and the Target Pattern.- 4.2.3 Phase Turbulence.- 4.3 The Phase Dynamics of Interfaces.- 4.4 Multiple Field Dynamics.- 5. Foundations of Reduction Theory.- 5.1 Two Simple Examples.- 5.2 The Destabilization of Stationary Solutions.- 5.3 Foundations of the Amplitude Equation.- 5.4 The Introduction of Continuous Spatial Degrees of Freedom.- 5.4.1 The Hopf Bifurcation.- 5.4.2 The Turing Instability.- 5.5 Fundamentals of Phase Dynamics.- 5.5.1 Phase Dynamics in a Uniform Oscillating Field.- 5.5.2 Phase Dynamics for a System with Periodic Structure.- 5.5.3 Interface Dynamics in a Two-Dimensional Medium.- Supplement I: Dynamics of Coupled Oscillator Systems.- SI.1 The Phase Dynamics of a Collection of Oscillators.- SI.2 Synchronization Phenomena.- II. The Structure and Physics of Chaos.- 6. A Physical Approach to Chaos.- 6.1 The Phase Space Structure of Dissipative Dynamical Systems.- 6.2 The Phase Space Structure of Conservative Dynamical Systems.- 6.3 Orbital Instability and the Mixing Nature of Chaos.- 6.3.1 The Liapunov Number.- 6.3.2 The Expansion Rate of Nearby Orbits, ?1(Xt ).- 6.3.3 Mixing and Memory Loss.- 6.4 The Statistical Description of Chaos.- 6.4.1 The Statistical Stability of Chaos.- 6.4.2 Time Coarse-Graining and the Spectrum ?(?).- 6.4.3 The Statistical Structure of Chaos.- 7. Bifurcation Phenomena of Dissipative Dynamical Systems.- 7.1 Band Chaos of the Hénon Map.- 7.2 The Derivation of Several Low-Dimensional Maps.- 7.2.1 The Hénon Map.- 7.2.2 The Annulus Map.- 7.2.3 The Standard Map (J = 1).- 7.2.4 One-Dimensional Maps (J = 0).- 7.3 Bifurcations of the One-Dimensional Quadratic Map.- 7.3.1 2n-Bifurcations and 2n-Band Bifurcations.- 7.3.2 The Self-Similarity and Renormalization Transformation of 2n-Bifurcations.- 7.3.3 The Similarity of 2n-Band Bifurcations.- 7.4 Bifurcations of the One-Dimensional Circle Map.- 7.4.1 Phase-Locked Band Chaos.- 7.4.2 Phase-Unlocked Fully Extended Chaos.- 8. The Statistical Physics of Aperiodic Motion.- 8.1 The Statistical Structure Functions of the Coarse-Grained Orbital Expansion Rate.- 8.1.1 The Baker Transformation.- 8.1.2 Attractor Destruction in the Quadratic Map.- 8.1.3 Attractor Merging in the Circle Map.- 8.1.4 Bifurcations of the Hénon Map.- 8.1.5 The Slopes s? and sß of ?(?).- 8.2 The Singularity Spectrum f(?).- 8.2.1 The Multifractal Dimension D(q).- 8.2.2 Partial Local Dimensions ?1(X) and ?2(X).- 8.2.3 f (?) Spectra of Critical Attractors.- 8.3 Theory Regarding the Slope of ?(?).- 8.3.1 The Slope s? Due to the Folding of Wu for Tangency Structure.- 8.3.2 The Slope sß Resulting from Collision with the Saddle S.- 8.4 The Relation Between f (?) and ?(?).- 8.4.1 The Linear Segment of f (?) Resulting from the Folding of Wu in the Presence of Tangency Structure.- 8.4.2 The Linear Segment of f (?) Caused by Bifurcation.- 9 Chaotic Bifurcations and Critical Phenomena.- 9.1 Crisis and Energy Dissipation in the Forced Pendulum.- 9.1.1 The Slope s? Induced by the Cantor Repellor.- 9.1.2 The Spectrum ?(W) of the Energy Dissipation Rate.- 9.1.3 The Formation of the Attractor Form in Figure 6.1.- 9.2 Fully-Extended Chaos That Exists After Attractor Merging.- 9.2.1 Attractor Merging in the Annulus Map.- 9.2.2 Attractor Merging in the Forced Pendulum.- 9.3 Critical Phenomena and Dynamical Similarity of Chaos.- 9.3.1 The Self-Similar Time Series of Critical Attractors.- 9.3.2 The Algebraic Structure Functions of the Critical Attractor.- 9.3.3 The Internal Similarity of Bands for the Spectrum ?(?).- 9.3.4 The Form Characterizing the Disappearance of Two-Dimensional Fractality.- 10. Mixing and Diffusion in Chaos of Conservative Systems.- 10.1 The Dynamical Self-Similarity of the Last KAM Torus.- 10.1.1 The Self-Similar Fm Time Series.- 10.1.2 The Symmetric Spectrum ?ß(ß).- 10.2 The Mixing of Widespread Chaos.- 10.2.1 The Form of ?(?) and the Breaking of Time-Reversal Symmetry.- 10.2.2 The Appearance of Anomalous Scaling Laws for Mixing.- 10.3 Anomalous Diffusion Due to Islands of Accelerator Mode Tori.- 10.3.1 Accelerator Mode Periodic Orbits.- 10.3.2 Long-Time Velocity Correlation.- 10.3.3 The Anomalous Nature of the Statistical Structure of the Coarse-Grained Velocity.- 10.4 Diffusion and Mixing of Fluids as a Result of Oscillation of Laminar Flow.- 10.4.1 Islands of Accelerator Mode Tori Existing Within Turnstiles.- 10.4.2 Anomalous Mixing Due to Long-Time Correlation.- Supplement II: On the Structure of Chaos.- SII.1 On-Off Intermittency.- SII.2 Anomalous Diffusion Induced by an Externally Applied Force.- SII.3 Transport Coefficients and the Liapunov Spectrum.- Summary of Part II.- A. Appendix.- A.1 Periodic Points of Conservative Maps and Their Neighborhoods.- A.2 Variance and the Time Correlation Function.- A.3 The Cantor Repellor of Intermittent Chaos.

O autorze

Dissipative Structures and Chaos is a far reaching description of the important field of Chaos Theory. This textbook, consisting of two parts, is written for graduate and postgraduate students as well as for scientists. The first part describes the phenomena leading to chaotic behaviour. The structure of chaotic systems is treated in the second part of the book.

Blurb

This monograph consists of two parts and gives an approach to the physics of open nonequilibrium systems. Part I derives the phenomena of dissipative structures on the basis of reduced evolution equations and includes Bénard convection and Belousov-Zhabotinskii chemical reactions. Part II discusses the physics and structures of chaos. While presenting a construction of the statistical physics of chaos, the authors unify the geometrical and statistical descriptions of dynamical systems. The shape of chaotic attractors is characterized, as are the mixing and diffusion of chaotic orbits and the fluctuation of energy dissipation exhibited by chaotic systems.