Applied Symbolic Dynamics and Chaos

The following sections are included:

• Contents

• Preface

The following sections are included:

• Dynamical Systems
  • Phase Space and Orbits
  • Parameters and Bifurcation of Dynamical Behavior
  • Examples of Dynamical Systems

• Symbolic Dynamics as Coarse-Grained Description of Dynamics
  • Fine-Grained and Coarse-Grained Descriptions
  • Symbolic Dynamics as the Simplest Dynamics

• Abstract versus Applied Symbolic Dynamics
  • Abstract Symbolic Dynamics
  • Applied Symbolic Dynamics

• Literature on Symbolic Dynamics

The following sections are included:

• Symbolic Sequences in Unimodal Maps
  • Numerical Orbit and Symbolic Sequence
  • Symbolic Sequence and Functional Composition
  • The Word-Lifting Technique

• The Quadratic Map
  • An Over-Simplified Population Model
  • Bifurcation Diagram of the Quadratic Map
  • Dark Lines in the Bifurcation Diagram

• Ordering of Symbolic Sequences and the Admissibility Condition
  • Property of Monotone Functions
  • The Ordering Rule
  • Dynamical Invariant Range and Kneading Sequence
  • The Admissibility Condition

• The Periodic Window Theorem
  • Periodic Window Theorem
  • Construction of Median Words
  • The MSS Table of Kneading Sequences
  • Nomenclature of Unstable Periodic Orbits

• Composition Rules
  • The *-Composition
  • Generalized Composition Rule
  • Proof of the Generalized Composition Rule
  • Applications of the Generalized Composition Rule
  • Further Remarks on Composition Rules

• Coarse-Grained Chaos
  • Chaos in the Surjective Unimodal Map
  • Chaos in ρλ^∞ Maps

• Topological Entropy

• Piecewise Linear Maps and Metric Representation of Symbolic Sequences
  • The Tent Map and Shift Map
  • The λ-Expansion of Real Numbers
  • Characteristic Function of the Kneading Sequence
  • Mapping of Subintervals and the Stefan Matrix
  • Markov Partitions and Generating Partitions
  • Metric Representation of Symbolic Sequences
  • Piecewise Linear Expanding Map

The following sections are included:

• General Discussion
  • The Ordering Rule
  • Admissibility and Compatibility of Kneading Sequences

• The Antisymmetric Cubic Map
  • Symbolic Sequences and Their Ordering
  • Admissibility Conditions
  • Generation of Superstable Median Words

• Symmetry Breaking and Restoration
  • Symmetry Breaking of Symmetric Orbits
  • Analysis of Symmetry Restoration

• The Gap Map
  • The Kneading Plane
  • Contacts of Even-Odd Type
  • Self-similar Structure in the Kneading Plane
  • Criterion for Topological Chaos

• The Lorenz-Like Map
  • Ordering Rule and Admissibility Conditions
  • Construction of the Kneading Plane
  • Contacts and Intersections
  • Farey and Doubling Transformations

• General Cubic Maps
  • Skeleton, Bones and Joints in Kneading Plane
  • The Construction of the Kneading Plane
  • The *-Composition Rules
  • The (−,+,−) Type Cubic Map

• The Sine-Square Map
  • Symbolic Sequences and Word-Lifting Technique
  • Ordering Rule and Admissibility Conditions
  • Generation of Kneading Sequences
  • Joints and Bones in the Kneading Plane
  • Skeleton of Superstable Orbits and Existence of Topological Chaos

• The Lorenz-Sparrow Maps
  • Ordering and Admissibility of Symbolic Sequences
  • Generation of Compatible Kneading Pairs
  • Generation of Admissible Sequences for a Given Kneading Pair
  • Metric Representation of Symbolic Sequences
  • One-Parameter Limits of Lorenz-Sparrow Maps

• Piecewise Linear Maps
  • Piecewise Linear Maps with Multiple Critical Points
  • Kneading Determinants

The following sections are included:

• The Physics of Linear and Nonlinear Oscillators

• Circle Maps and Their Lifts
  • The Rigid Rotation — Bare Circle Map
  • The Sine-Circle Map
  • Lift of Circle Maps
  • Rotation Number and Rotation Interval
  • Arnold Tongues in the Parameter Plane

• Continued Fractions and Farey Tree
  • Farey Tree: Rational Fraction Representation
  • Farey Tree: Continued Fraction Representation
  • Farey Tree: Farey Addresses and Farey Matrices
  • More on Continued Fractions and Farey Representations
  • Farey Tree: Symbolic Representation

• Farey Transformations and Well-Ordered Orbits
  • Well-Ordered Symbolic Sequences
  • Farey Transformations as Composition Rules
  • Extreme Property of Well-Ordered Periodic Sequences
  • Generation of R^o_mak, and L^o_min

• Circle Map with Non-Monotone Lift
  • Symbolic Sequences and Their Continuous Transformations
  • Ordering Rule and Admissibility Condition
  • Existence of Well-Ordered Symbolic Sequences
  • The Farey Transformations
  • Existence of Symbolic Sequence without Rotation Number

• Kneading Plane of Circle Maps
  • Arnold Tongue with Rotation Number ½
  • Doubly Superstable Kneading Sequences: Joints and Bones
  • Generation of Kneading Sequences K_g and K_s
  • Construction of the Kneading Plane

• Piecewise Linear Circle Maps and Topological Entropy
  • The Sawtooth Circle Map
  • Circle Map with Given Kneading Sequences
  • Kneading Determinant and Topological Entropy
  • Construction of a Map from a Given Kneading Sequence
  • Rotation Interval and Well-Ordered Periodic Sequences

The following sections are included:

• General Discussion
  • Bi-Infinite Symbolic Sequences
  • Decomposition of the Phase Plane
  • Tangencies and Admissibility Conditions
  • Admissibility Conditions in Symbolic Plane

• Invariant Manifolds and Dynamical Foliations of Phase Plane
  • Stable and Unstable Invariant Manifolds
  • Dynamical Foliations of the Phase Plane
  • Summary and Discussion

• The Tél Map
  • Forward and Backward Symbolic Sequences
  • Dynamical Foliations of Phase Space and Their Ordering
  • Forbidden and Allowed Zones in Symbolic Plane
  • The Admissibility Conditions
  • Summary

• The Lozi Map
  • Forward and Backward Symbolic Sequences
  • Dynamical Foliations of the Phase Space
  • Ordering of the Forward and Backward Foliations
  • Allowed and Forbidden Zones in Symbolic Plane
  • Discussion of the Admissibility Condition

• The Hénon Map
  • Fixed Points and Their Stability
  • Determination of Partition Lines in Phase Plane
  • Hénon-Type Symbolic Dynamics
  • Symbolic Analysis at Typical Parameter Values

• The Dissipative Standard Map
  • Dynamical Foliations of the Phase Plane
  • Ordering of Symbolic Sequences
  • Symbolic Plane and Admissibility of Symbolic Sequences

• The Stadium Billiard Problem
  • A Coding Based on Lifting
  • Relation to Other Codings
  • The Half-Stadium
  • Summary

The following sections are included:

• General Discussion
  • Three Types of ODEs
  • On Numerical Integration of Differential Equations
  • Numerical Calculation of the Poincaré Maps

• The Periodically Forced Brusselator
  • The Brusselator Viewed from The Standard Map
  • Transition from Annular to Interval Dynamics
  • Symbolic Analysis of Interval Dynamics

• The Lorenz Equations
  • Summary of Known Properties
  • Construction of Poincaré and Return Maps
  • One-Dimensional Symbolic Dynamics Analysis
  • Symbolic Dynamics of the 2D Poincaré Maps
  • Stable Periodic Orbits
  • Concluding Remarks

• Summary of Other ODE Systems
  • The Driven Two-Well Duffing Equation
  • The NMR-Laser Model

The following sections are included:

• Periodic versus Chaotic Regimes
  • Stable Versus Unstable Periods in 1D Maps
  • Notations and Summary of Results
  • A Few Number Theory Notations and Functions

• Number of Periodic Orbits in a Class of One-Parameter Maps
  • Number of Admissible Words in Symbolic Dynamics
  • Number of Tangent and Period-Doubling Bifurcations
  • Recursion Formula for the Total Number of Periods
  • Symmetry Types of Periodic Sequences
  • Explicit Solutions to the Recurrence Relations
  • Finite Lambda Auto-Expansion of Real Numbers

• Other Aspects of the Counting Problem
  • The Number of Roots of the “Dark Line” Equation
  • Number of Saddle Nodes in Forming Smale Horseshoe
  • Number of Solutions of Renormalization Group Equations

• Counting Formulae for General Continuous Maps

• Number of Periods in Maps With Discontinuity
  • Number of Periods in the Gap Map
  • Number of Periods in the Lorenz-Like Map

• Summary of the Counting Problem

• Cycle Expansion for Topological Entropy

The following sections are included:

• Formal Languages and Their Complexity
  • Formal Language
  • Chomsky Hierarchy of Grammatical Complexity
  • The L-System

• Regular Language and Finite Automaton
  • Finite Automaton
  • Regular Language
  • Stefan Matrix as Transfer Function for Automaton

• Beyond Regular Languages
  • Feigenbaum and Generalized Feigenbaum Limiting Sets
  • Even and Odd Fibonacci Sequences
  • Odd Maximal Primitive Prefixes and Kneading Map
  • Even Maximal Primitive Prefixes and Distinct Excluded Blocks

• Summary of Results

The following sections are included:

• Knots and Links

• Knots and Links from Unimodal Maps

• Linking Numbers

• Discussion

The following sections are included:

• APPENDIX

• REFERENCES

• Index