Robust Chaos and Its Applications

The following sections are included:

• Preface

• Contents

The following sections are included:

• Poincaré and generalized Poincaré mappings

• Interval methods for calculating Poincaré mappings
  • Existence of periodic orbits
  • Interval arithmetic

• Smale horseshoe
  • Dynamics of the horseshoe map

• Symbolic dynamics
  • The method of fixed point index
    • Periodic points of the TS-map
    • Existence of semi-conjugacy

• Exercises

The following sections are included:

• Strange attractors
  • Concepts and definitions
  • Robust chaos
  • Domains of attraction

• Density and robustness of chaos

• Persistence and robustness of chaos

• Exercises

The following sections are included:

• Entropies
  • Lebesgue (volume) measure
  • Physical (or Sinai–Ruelle–Bowen) measure
  • Hausdorff dimension
  • The topological entropy
  • Lyapunov exponent

• Ergodic theory

• Statistical properties of chaotic attractors
  • Autocorrelation function (ACF)
  • Correlations

• Exercises

The following sections are included:

• The concept of structural stability
  • Conditions for structural stability
  • A proof of Anosov's theorem on structural stability of diffeomorphisms

• Exercises

The following sections are included:

• Transversality

• Invariant foliation

• Shadowing lemma
  • Homoclinic orbits and shadowing
  • Shilnikov criterion for the existence of chaos

• Exercises

The following sections are included:

• Hyperbolic dynamics
  • Concepts and definitions
  • Anosov diffeomorphisms and Anosov flows

• Anosov diffeomorphisms on the torus 𝕋_n
  • Anosov automorphisms
  • Structure of Anosov diffeomorphisms
  • Anosov torus 𝕋_n with a hyperbolic structure
  • Expanding maps
  • The Blaschke product
  • The Bernoulli map
  • The Arnold cat map

• Classification of strange attractors of dynamical systems

• Properties of hyperbolic chaotic attractors
  • Geodesic flows on compact smooth manifolds
  • The solenoid attractor
  • The Smale–Williams solenoid
  • Plykin attractor

• Proof of the hyperbolicity of the logistic map for μ > 4

• Generalized hyperbolic attractors

• Generating hyperbolic attractors

• Density of hyperbolicity and homoclinic bifurcations in arbitrary dimension

• Hyperbolicity tests
  • Numerical procedure
  • Testing hyperbolicity of the Hénon map
    • Detect whether the minimum angle tends to zero
    • Theoretical model for the fraction of nonhyperbolic parameter values
  • Testing hyperbolicity of the forced damped pendulum

• Uniform hyperbolicity test

• Exercises

The following sections are included:

• Modeling hyperbolic attractors
  • Modeling the Smale–Williams attractor
  • Testing hyperbolicity of system
  • Numerical verification of the hyperbolicity of system
  • Modeling the Arnold cat map
  • Modeling the Bernoulli map
  • Modeling Plykin's attractor

• Exercises

The following sections are included:

• Lorenz-type attractors

• The Lorenz system
  • Existence of Lorenz-type attractors
  • Geometric models of the Lorenz equation
    • Chaoticity of the geometric Lorenz attractor
    • Transitivity of the Lorenz attractor
    • Singular horseshoe
    • Lorenz-like families with criticalities
    • Topological classification of geometrical Lorenz attractors
  • Structure of the Lorenz attractor
    • Existence of short periodic orbits for the Lorenz system

• Expanding and contracting Lorenz attractors

• Wild strange attractors and pseudo-hyperbolicity

• Lorenz-type attractors realized in two-dimensional maps

• Exercises

The following sections are included:

• Robust chaos in the Lorenz-type attractors

• Robust chaos in Lorenz system

• Robust chaos in 2-D Lorenz-type attractors

• Exercises

The following sections are included:

• Quasi-attractors, concepts, and properties

• The Hénon map
  • Uniform hyperbolicity of the Hénon map
  • Hyperbolicity of Hénon-like maps
  • Hénon attractor is a quasi-attractor

• The Strelkova–Anishchenko map

• The Anishchenko–Astakhov oscillator

• Chua's circuit
  • Homoclinic and heteroclinic orbits
  • The geometric model

• Exercises

The following sections are included:

• Unimodal maps
  • S-unimodal maps
  • Relation between unimodality and hyperbolicity
  • Classification of unimodal maps of the interval
  • Collet–Eckmann maps
  • Statistical properties of unimodal maps

• The Barreto–Hunt–Grebogi–Yorke conjecture
  • Counter-examples to the Barreto–Hunt–Grebogi–Yorke conjecture
  • Robust chaos without the period-n-tupling scenario
  • The B-exponential map

• Border-collision bifurcation and robust chaos
  • Normal form for piecewise-smooth one-dimensional maps
  • Border-collision bifurcation scenarios
    • Scenario A. Persistent fixed point (nonbifurcation)
    • Scenario B. Border-collision pair bifurcation
    • Scenario C. border-crossing bifurcation
  • Robust chaos in one-dimensional singular maps

• Exercises

The following sections are included:

• Robust chaos in 2-D piecewise-smooth maps
  • Normal form for 2-D piecewise-smooth maps
  • Border-collision bifurcations and robust chaos
    • General classification
    • The possible types of fixed points of the normal form of map
  • Regions for nonrobust chaos
    • The case of positive determinants on both sides of the border
    • The case of negative determinants on both sides of the border
    • The case of negative determinant to the left of the border and positive determinant to the right of the border
    • The case of positive determinant to the left of the border and negative determinant to the right of the border
    • Stable fixed point leading to stable fixed point plus extraneous periodic orbits
  • Regions for robust chaos: undesirable and dangerous bifurcations
    • Border-collision pair bifurcation
    • Subcritical border-collision period doubling
    • Supercritical border-collision period doubling
    • Stable fixed point leading to chaos
  • Proof of unicity of orbits
  • Proof of robust chaos
    • Case 1
    • Case 2
    • Case 3
    • Example 1
    • Example 2

• Robust chaos in noninvertible piecewise-linear maps
  • Normal forms for two-dimensional noninvertible maps
  • Onset of chaos: Proof of robust chaos
    • Border-collision from stable fixed point to flip saddle
    • Two-piece invariant sets
    • Three-piece invariant sets
    • N-piece invariant sets

• Exercises

The following sections are included:

• Bibliography

• Index