1 Introduction

1.1 Dynamical Systems

1.1.1 Phase Space and Orbits

1.1.2 Parameters and Bifurcation of Dynamical Behavior

1.1.3 Examples of Dynamical Systems

1.2 Symbolic Dynamics as Coarse-Grained Description of Dynamics

1.2.1 Fine-Grained and Coarse-Grained Descriptions

1.2.2 Symbolic Dynamics as the Simplest Dynamics

1.3 Abstract versus Applied Symbolic Dynamics

1.3.1 Abstract Symbolic Dynamics

1.3.2 Applied Symbolic Dynamics

1.4 Literature on Symbolic Dynamics

2 Symbolic Dynamics of Unimodal Maps

2.1 Symbolic Sequences in Unimodal Maps

2.1.1 Numerical Orbit and Symbolic Sequence

2.1.2 Symbolic Sequence and Functional Composition

2.1.3 The Word-Lifting Technique

2.2 The Quadratic Map

2.2.1 An Over-Simplified Population Model

2.2.2 Bifurcation Diagram of the Quadratic Map

2.2.3 Dark Lines in the Bifurcation Diagram

2.3 Ordering of Symbolic Sequences and the Admissibility Condition

2.3.1 Property of Monotone Functions

2.3.2 The Ordering Rule

2.3.3 Dynamical Invariant Range and Kneading Sequence

2.3.4 The Admissibility Condition

2.4 The Periodic Window Theorem

2.4.1 The Periodic Window Theorem

2.4.2 Construction of Median Words

2.4.3 The MSS Table of Kneading Sequences

2.4.4 Nomenclature of Unstable Periodic Orbits

2.5 Composition Rules

2.5.1 The*-Composition

2.5.2 Generalized Composition Rule

2.5.3 Proof of the Generalized Composition Rule

2.5.4 Applications of the Generalized Composition Rule

2.5.5 Further Remarks on Composition Rules

2.6 Coarse-Grained Chaos

2.6.1 Chaos in the Surjective Unimodal Map

2.6.2 Chaos in ρλ∞Maps

2.7 Topological Entropy

2.8 Piecewise Linear Maps and Metric Representation of Symbolic Sequences

2.8.1 The Tent Map and Shift Map

2.8.2 Theλ-Expansion of Real Numbers

2.8.3 Characteristic Function of the Kneading Sequence

2.8.4 Mapping of Subintervals and the Stefan Matrix

2.8.5 Markov Partitions and Generating Partitions

2.8.6 Metric Representation of Symbolic Sequences

2.8.7 Piecewise Linear Expanding Map

3 Maps with Multiple Critical Points

3.1 General Discussion

3.1.1 The Ordering Rule

3.1.2 Construction of a Map from a Given Kneading Sequence

3.2 The Antisymmetric Cubic Map

3.2.1 Symbolic Sequences and Their Ordering

3.2.2 Admissibility Conditions

3.2.3 Generation of Superstable Median Words

3.3 Symmetry Breaking and Restoration

3.3.1 Symmetry Breaking of Symmetric Orbits

3.3.2 Analysis of Symmetry Restoration

3.4 The Gap Map

3.4.1 The Kneading Plane

3.4.2 Contacts of Even-Odd Type

3.4.3 Self-Similar Structure in the Kneading Plane

3.4.4 Criterion for Topological Chaos

3.5 The Lorenz-Like Map

3.5.1 Ordering Rule and Admissibility Conditions

3.5.2 Construction of the Kneading Plane

3.5.3 Contacts and Intersections

3.5.4 Farey and Doubling Transformations

3.6 General Cubic Maps

3.6.1 Skeleton，Bones and Joints in Kneading Plane

3.6.2 The Construction of the Kneading Plane

3.6.3 The*-Composition Rules

3.6.4 The Down-Up-Down Type Cubic Map

3.7 The Sine-Square Map

3.7.1 Symbolic Sequences and Word-Lifting Technique

3.7.2 Ordering Rule and Admissibility Conditions

3.7.3 Generation of Kneading Sequences

3.7.4 Joints and Bones in the Kneading Plane

3.7.5 Skeleton of Superstable Orbits and Existence of Topological Chaos

3.8 The Lorenz-Sparrow Maps

3.8.1 Ordering and Admissibility of Symbolic Sequences

3.8.2 Generation of Compatible Kneading Pairs

3.8.3 Generation of Admissible Sequences for a Given Kneading Pair

3.8.4 Metric Representation of Symbolic Sequences

3.8.5 One-Parameter Limits of Lorenz-Sparrow Maps

3.9 Piecewise Linear Maps

3.9.1 Piecewise Linear Maps with Multiple Critical Points

3.9.2 Kneading Determinants

4 Symbolic Dynamics of Circle Maps

4.1 The Physics of Linear and Nonlinear Oscillators

4.2 Circle Maps and Their Lifts

4.2.1 The Rigid Rotation—Bare Circle Map

4.2.2 The Sine-Circle Map

4.2.3 Lift of Circle Maps

4.2.4 Rotation Number and Rotation Interval

4.2.5 Arnold Tongues in the Parameter Plane

4.3 Continued Fractions and Farey Tree

4.3.1 Farey Tree：Rational Fraction Representation

4.3.2 Farey Tree：Continued Fraction Representation

4.3.3 Farey Tree：Farey Addresses and Farey Matrices

4.3.4 More on Continued Fractions and Farey Representations

4.3.5 Farey Tree：Symbolic Representation

4.4 Farey Transformations and Well-Ordered Orbits

4.4.1 Well-Ordered Symbolic Sequences

4.4.2 Farey Transformations as Composition Rules

4.4.3 Extreme Property of Well-Ordered Periodic Sequences

4.4.4 Generation of R°max and L°min

4.5 Circle Map with Non-Monotone Lift

4.5.1 Symbolic Sequences and Their Continuous Transformations

4.5.2 Ordering Rule and Admissibility Condition

4.5.3 Existence of Well-Ordered Symbolic Sequences

4.5.4 The Farey Transformations

4.5.5 Existence of Symbolic Sequence without Rotation Number

4.6 Kneading Plane of Circle Maps

4.6.1 Arnold Tongue with Rotation Number 1/2

4.6.2 Doubly Superstable Kneading Sequences：Joints and Bones

4.6.3 Generation of Kneading Sequences Kg and Ks

4.6.4 Construction of the Kneading Plane

4.7 Piecewise Linear Circle Maps and Topological Entropy

4.7.1 The Sawtooth Circle Map

4.7.2 Circle Map with Given Kneading Sequences

4.7.3 Kneading Determinant and Topological Entropy

4.7.4 Construction of a Map from a Given Kneading Sequence

4.7.5 Rotation Interval and Well-Ordered Periodic Sequences

5 Symbolic Dynamics of Two-Dimensional Maps

5.1 General Discussion

5.1.1 Bi-Infinite Symbolic Sequences

5.1.2 Decomposition of the Phase Plane

5.1.3 Tangencies and Admissibility Conditions

5.1.4 Admissibility Conditions in Symbolic Plane

5.2 Invariant Manifolds and Dynamical Foliations of Phase Plane

5.2.1 Stable and Unstable Invariant Manifolds

5.2.2 Dynamical Foliations of the Phase Plane

5.2.3 Summary and Discussion

5.3 The Tél Map

5.3.1 Forward and Backward Symbolic Sequences

5.3.2 Dynamical Foliations of Phase Space and Their Ordering

5.3.3 Forbidden and Allowed Zones in Symbolic Plane

5.3.4 The Admissibility Conditions

5.3.5 Summary

5.4 The Lozi Map

5.4.1 Forward and Backward Symbolic Sequences

5.4.2 Dynamical Foliations of the Phase Space

5.4.3 Ordering of the Forward and Backward Foliations

5.4.4 Allowed and Forbidden Zones in the Symbolic Plane

5.4.5 Discussion of the Admissibility Condition

5.5 The Hénon Map

5.5.1 Fixed Points and Their Stability

5.5.2 Determination of Partition Lines in Phase Plane

5.5.3 Hénon-Type Symbolic Dynamics

5.5.4 Symbolic Analysis at Typical Parameter Values

5.5.5 Discussion

5.6 The Dissipative Standard Map

5.6.1 Dynamical Foliations of the Phase Plane

5.6.2 Ordering of Symbolic Sequences

5.6.3 Symbolic Plane and Admissibility of Symbolic Sequences

5.7 The Stadium Billiard Problem

5.7.1 A Coding Based on Lifting

5.7.2 Relation to Other Codings

5.7.3 The Half-Stadium

5.7.4 Summary

6 Application to Ordinary Differential Equations

6.1 General Discussion

6.1.1 Three Types of ODEs

6.1.2 On Numerical Integration of Differential Equations

6.1.3 Numerical Calculation of the Poincaré Maps

6.2 The Periodically Forced Brusselator

6.2.1 The Brusselator Viewed from The Standard Map

6.2.2 Transition from Annular to Interval Dynamics

6.2.3 Symbolic Analysis of Interval Dynamics

6.3 The Lorenz Equations

6.3.1 Summary of Known Properties

6.3.2 Construction of Poincaré and Return Maps

6.3.3 One-Dimensional Symbolic Dynamics Analysis

6.3.4 Symbolic Dynamics of the 2D Poincaré Maps

6.3.5 Stable Periodic Orbits

6.3.6 Concluding Remarks

6.4 Summary of Other ODE Systems

6.4.1 The Driven Two-Well Duffing Equation

6.4.2 The NMR-Laser Model

7 Counting the Number of Periodic Orbits

7.1 Periodic versus Chaotic Regimes

7.1.1 Stable Versus Unstable Periods in 1D Maps

7.1.2 Notations and Summary of Results

7.1.3 A Few Number Theory Notations and Functions

7.2 Number of Periodic Orbits in a Class of One-Parameter Maps

7.2.1 Number of Admissible Words in Symbolic Dynamics

7.2.2 Number of Tangent and Period-Doubling Bifurcations

7.2.3 Recursion Formula for the Total Number of Periods

7.2.4 Symmetry Types of Periodic Sequences

7.2.5 Explicit Solutions to the Recurrence Relations

7.2.6 Finite Lambda Auto-Expansion of Real Numbers

7.3 Other Aspects of the Counting Problem

7.3.1 The Number of Roots of the“Dark Line”Equation

7.3.2 Number of Saddle Nodes in Forming Smale Horseshoe

7.3.3 Number of Solutions of Renormalization Group Equations

7.4 Counting Formulae for General Continuous Maps

7.5 Number of Periods in Maps With Discontinuity

7.5.1 Number of Periods in the Gap Map

7.5.2 Number of Periods in the Lorenz-Like Map

7.6 Summary of the Counting Problem

7.7 Cycle Expansion for Topological Entropy

8 Symbolic Dynamics and Grammatical Complexity

8.1 Formal Languages and Their Complexity

8.1.1 Formal Language

8.1.2 Chomsky Hierarchy of Grammatical Complexity

8.1.3 The L-System

8.2 Regular Language and Finite Automaton

8.2.1 Finite Automaton

8.2.2 Regular Language

8.2.3 Stefan Matrix as Transfer Function for Automaton

8.3 Beyond Regular Languages

8.3.1 Feigenbaum and Generalized Feigenbaum Limiting Sets

8.3.2 Even and Odd Fibonacci Sequences

8.3.3 Odd Maximal Primitive Prefixes and Kneading Map

8.3.4 Even Maximal Primitive Prefixes and Distinct Excluded Blocks

8.4 Summary of Results

9 Symbolic Dynamics and Knot Theory

9.1 Knots and Links

9.2 Knots and Links from Unimodal Maps

9.3 Linking Numbers

9.4 Discussion

10 Appendix

A.1 Program to Generate Admissible Sequences

A.2 Program to Draw Dynamical Foliations of a Two-Dimensional Map

A.3 A Greedy Program for Determining Partition Line

References

R.1 Books

R.2 Papers

Index