Dissipative Structures and Chaos

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.