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Chaos, Dynamics, and Fractals

Book Description

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.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Introduction
  9. 1. Flows in phase space
    1. 1.1 Determinism, phase flows, and Liouville’s theorem
    2. 1.2 Equilibria, linear stability, and limit cycles
    3. 1.3 Change of stability (bifurcations)
    4. 1.4 Periodically driven systems and stroboscopic maps
    5. 1.5 Continuous groups of transformations as phase space flows
  10. 2. Introduction to deterministic chaos
    1. 2.1 The Lorenz model, the Lorenz plot, and the binary tent map
    2. 2.2 Local exponential instability of nearby orbits: the positive Liapunov exponent
    3. 2.3 The Frobenius–Peron equation (invariant densities)
    4. 2.4 Simple examples of fully developed chaos for maps of the interval
    5. 2.5 Maps that are conjugate under differentiable coordinate transformations
    6. 2.6 Computation of nonperiodic chaotic orbits at fully developed chaos
    7. 2.7 Is the idea of randomness necessary in natural science?
  11. 3. Conservative dynamical systems
    1. 3.1 Integrable conservative systems: symmetry, invariance, conservation laws, and motion on invariant tori in phase space
    2. 3.2 The Hénon–Heiles model: evidence for bifurcations from integrable to chaotic behavior
    3. 3.3 Perturbed twist maps: nearly integrable conservative systems
    4. 3.4 Mixing and ergodicity: the approach to statistical equilibrium
    5. 3.5 The bakers’ transformation
    6. 3.6 Computation of chaotic orbits for an area-preserving map
    7. Appendix 3.A Generating functions for canonical transformations
    8. Appendix 3.B Systems in involution
  12. 4. Fractals and fragmentation in phase space
    1. 4.1 Introduction to fractals
    2. 4.2 Geometrically selfsimilar fractals
    3. 4.3 The dissipative bakers’ transformation: a model ‘strange’ attractor
    4. 4.4 The symmetric tent map: a model ‘strange’ repeller
    5. 4.5 The devil’s staircase: arithmetic on the Cantor set
    6. 4.6 Generalized dimensions and the coarsegraining of phase space
    7. 4.7 Computation of chaotic orbits on a fractal
  13. 5. The way to chaos by instability of quasiperiodic orbits
    1. 5.1 From limit cycles to tori to chaos
    2. 5.2 Periodically driven systems and circle maps
    3. 5.3 Arnol’d tongues and the devil’s staircase
    4. 5.4 Scaling laws and renormalization group equations
    5. 5.5 The Farey tree
  14. 6. The way to chaos by period doubling
    1. 6.1 Universality at transitions to chaos
    2. 6.2 Instability of periodic orbits by period doubling
    3. 6.3 Universal scaling for noninvertible quadratic maps of the interval
  15. 7. Introduction to multifractals
    1. 7.1 Incomplete but optimal information: the natural coarsegraining of phase space
    2. 7.2 The f(α)-spectrum
    3. 7.3 The asymmetric tent map and the two-scale Cantor set (f(α) and entropy)
    4. 7.4 Multifractals at the borderlines of chaos
  16. 8. Statistical mechanics on symbol sequences
    1. 8.1 Introduction to statistical mechanics
    2. 8.2 Introduction to symbolic dynamics
    3. 8.3 The transfer matrix method
    4. 8.4 What is the temperature of chaotic motion on a fractal?
    5. 8.5 [f(bar)](<α>) as a thermodynamic prediction in the canonical ensemble
    6. 8.6 Phase transitions
  17. 9. Universal chaotic dynamics
    1. 9.1 Invariant probability distributions from chaos
    2. 9.2 Symbol sequence universality
    3. 9.3 Universal deterministic statistical independence
    4. 9.4 Deterministic noise
    5. 9.5 Trees of higher order and incomplete trees
  18. 10. Intermittence in fluid turbulence
    1. 10.1 Fluid turbulence in open flows
    2. 10.2 Scale invariance and broken symmetry
    3. 10.3 Kolmogorov’s 1941 theory (K41 theory)
    4. 10.4 The β-model of the inertial range cascade: a model of spatial intermittence
    5. 10.5 Multifractal models of intermittence
    6. 10.6 Intermittence of energy dissipation at the small scales of turbulence
  19. 11. From flows to automata: chaotic systems as completely deterministic machines
    1. 11.1 The arithmetic of deterministic chaos
    2. 11.2 The shadowing lemma
    3. 11.3 The problem with the continuum and ‘measure-one’
    4. 11.4 Why measure-one behavior may be hard to find
    5. 11.5 The idea of the universal computer
  20. Bibliography
  21. Index