Applied Symbolic Dynamics and Chaos

by Hao, Bai-Lin; Zheng, Wei-Mou

• ISBN13: 9789810235123
• ISBN10: 9810235127
• Format: Hardcover
• Copyright: 1998-09-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Attempts to span the gap between the mathematics- oriented literature on symbolic dynamics & making symbolic dynamics a practical tool in studying chaotic dynamics, both on computer & in laboratories. The abstract formulation of dynamical systems has kept many practitioners of physical sciences & engineering from appreciating its simplicity, beauty, & power.

Table of Contents

Preface	xiii
1 Introduction	1	(12)
1.1 Dynamical Systems	2	(3)
1.1.1 Phase Space and Orbits	2	(1)
1.1.2 Parameters and Bifurcation of Dynamical Behavior	2	(1)
1.1.3 Examples of Dynamical Systems	3	(2)
1.2 Symbolic dynamics as Coarse-Grained Description of Dynamics	5	(3)
1.2.1 Fine- Grained and Coarse-Grained Descriptions	6	(1)
1.2.2 Symbolic Dynamics as the Simplest Dynamics	6	(2)
1.3 Abstract versus Applied Symbolic Dynamics	8	(3)
1.3.1 Abstract Symbolic Dynamics	8	(1)
1.3.2 Applied Symbolic Dynamics	9	(2)
1.4 Literature on Symbolic Dynamics	11	(2)
2 Symbolic Dynamics of Unimodal Maps	13	(90)
2.1 Symbolic Sequences in Unimodal Maps	15	(9)
2.1.1 Numerical Orbit and Symbolic Sequence	15	(6)
2.1.2 Symbolic Sequence and Functional Composition	21	(1)
2.1.3 The Word-Lifting Technique	22	(2)
2.2 The Quadratic Map	24	(13)
2.2.1 An Over- Simplifed Population Model	24	(2)
2.2.2 Bifurcation Diagram of the Quadratic Map	26	(4)
2.2.3 Dark Lines in the Bifurcation Diagram	30	(7)
2.3 Ordering of Symbolic Sequences and the Admissibility Condition	37	(8)
2.3.1 Property of Monotone Functions	38	(1)
2.3.2 The Ordering Rule	38	(4)
2.3.3 Dynamical Invariant Range and Kneading Sequence	42	(1)
2.3.4 The Admissibility Condition	43	(2)
2.4 The Periodic Window Theorem	45	(8)
2.4.1 Periodic Window Theorem	46	(3)
2.4.2 Construction of Median Words	49	(1)
2.4.3 The MSS Table of Kneading Sequences	50	(2)
2.4.4 Nomenclature of Unstable Periodic Orbits	52	(1)
2.5 Composition Rules	53	(14)
2.5.1 The *-Composition	53	(4)
2.5.2 Generalized Composition Rule	57	(3)
2.5.3 Proof of the Generalized Composition Rule	60	(2)
2.5.4 Applications of the Generalized Composition Rule	62	(3)
2.5.5 Further Remarks on Composition Rules	65	(2)
2.6 Coarse-Grained Chaos	67	(15)
2.6.1 Chaos in the Surjective Unimodal Map	68	(6)
2.6.2 Chaos in p Lambda(xxx) Maps	74	(8)
2.7 Topological Entropy	82	(1)
2.8 Piecewise Linear Maps and Metric Representation of Symbolic Sequences	83	(20)
2.8.1 The Tent Map and Shift Map	84	(1)
2.8.2 The Lambda-Expansion of Real Numbers	85	(1)
2.8.3 Characteristic Function of the Kneading Sequence	86	(1)
2.8.4 Mapping of Subintervals and the Stefan Matrix	86	(7)
2.8.5 Markov Partitions and Generating Partitions	93	(3)
2.8.6 Metric Representation of Symbolic Sequences	96	(4)
2.8.7 Piecewise Linear Expanding Map	100	(3)
3 Maps with Multiple Critical Points	103	(74)
3.1 General Discussion	104	(2)
3.1.1 The Ordering Rule	105	(1)
3.1.2 Admissibility and Compatibility of Kneading Sequences	106	(1)
3.2 The Antisymmetric Cubic Map	106	(12)
3.2.1 Symbolic Sequences and Their Ordering	109	(1)
3.2.2 Admissibility Conditions	110	(2)
3.2.3 Generation of Superstable Median Words	112	(6)
3.3 Symmetry Breaking and Restoration	118	(7)
3.3.1 Symmetry Breaking of Symmetric Orbits	120	(2)
3.3.2 Analysis of Symmetry Restoration	122	(3)
3.4 The Gap Map	125	(13)
3.4.1 The Kneading Plane	127	(4)
3.4.2 Contacts of Even-Odd Type	131	(1)
3.4.3 Self-Similar Structure in the Kneading Plane	132	(3)
3.4.4 Criterion for Topological Chaos	135	(3)
3.5 The Lorenz-Like Map	138	(4)
3.5.1 Ordering Rule and Admissibility Conditions	139	(1)
3.5.2 Construction of the Kneading Plane	139	(1)
3.5.3 Contacts and Intersections	140	(1)
3.5.4 Farey and Doubling Transformations	141	(1)
3.6 General Cubic Maps	142	(14)
3.6.1 Skeleton, Bones and Joints in Kneading Plane	145	(2)
3.6.2 The Construction of the Kneading Plane	147	(4)
3.6.3 The *-Composition Rules	151	(1)
3.6.4 The (-,+,-) Type Cubic Map	152	(4)
3.7 The Sine-Square Map	156	(10)
3.7.1 Symbolic Sequences and Word-Lifting Technique	157	(2)
3.7.2 Ordering Rule and Admissibility Conditions	159	(1)
3.7.3 Generation of Kneading Sequences	160	(1)
3.7.4 Joints and Bones in the Kneading Plane	161	(4)
3.7.5 Skeleton of Superstable Orbits and Existence of Topological Chaos	165	(1)
3.8 The Lorenz-Sparrow Maps	166	(8)
3.8.1 Ordering and Admissibility of Symbolic Sequences	167	(2)
3.8.2 Generation of Compatible Kneading Pairs	169	(1)
3.8.3 Generation of Admissible Sequences for Given Kneading Pair	170	(2)
3.8.4 Metric Representation of Symbolic Sequences	172	(1)
3.8.5 One-Parameter Limits of Lorenz-Sparrow Maps	173	(1)
3.9 Piecewise Linear Maps	174	(3)
3.9.1 Piecewise Linear Maps with Multiple Critical Points	174	(1)
3.9.2 Kneading Determinants	175	(2)
4 Symbolic Dynamics of Circle Maps	177	(48)
4.1 The Physics of Linear and Nonlinear Oscillators	178	(1)
4.2 Circle Maps and Their Lifts	179	(8)
4.2.1 The Rigid Rotation-Bare Circle Map	180	(2)
4.2.2 The Sine-Circle Map	182	(1)
4.2.3 Lift of Circle Maps	183	(1)
4.2.4 Rotation Number and Rotation Interval	184	(2)
4.2.5 Arnold Tongues in the Parameter Plane	186	(1)
4.3 Continued Fractions and Farey Tree	187	(13)
4.3.1 Farey Tree: Rational Fraction Representation	187	(1)
4.3.2 Farey Tree: Continued Fraction Representation	188	(3)
4.3.3 Farey Tree: Farey Addresses and Farey Matrices	191	(2)
4.3.4 More on Continued Fraction and Farey Representations	193	(4)
4.3.5 Farey Tree: Symbolic Representation	197	(3)
4.4 Farey Transformations and Well-Ordered Orbits	200	(7)
4.4.1 Well-Ordered Symbolic Sequences	201	(1)
4.4.2 Farey Transformations as Composition Rules	201	(1)
4.3.3 Extreme Property of Well-Ordered Periodic Sequences	202	(3)
4.4.4 Generation of R(degree)(max) and L(degree)(min)	205	(2)
4.5 Circle Map with Non-Monotone Lift	207	(5)
4.5.1 Symbolic Sequences and Their Continuous Transformations	207	(1)
4.5.2 Ordering Rule and Admissibility Condition	208	(1)
4.5.3 Existence of Well-Ordered Symbolic Sequences	209	(1)
4.5.4 The Farey Transformations	210	(1)
4.5.5 Existence of Symbolic Sequence without Rotation Number	211	(1)
4.6 Kneading Plane of Circle Maps	212	(6)
4.6.1 Arnold Tongue with Rotation Number 1/2	212	(1)
4.6.2 Doubly Superstable Kneading Sequences: Joints and Bones	213	(2)
4.6.3 Generation of Kneading Sequences K(g) and K(s)	215	(2)
4.6.4 Construction of the Kneading Plane	217	(1)
4.7 Piecewise Linear Circle Maps and Topological Entropy	218	(7)
4.7.1 The Sawtooth Circle Map	218	(1)
4.7.2 Circle Map with Given Kneading Sequences	219	(2)
4.7.3 Kneading Determinant and Topological Entropy	221	(1)
4.7.4 Construction of a Map from a Given Kneading Sequences	222	(1)
4.7.5 Rotation Interval and Well-Ordered Periodic Sequences	222	(3)
5 Symbolic Dynamics of Two-Dimensional Maps	225	(78)
5.1 General Discussion	227	(6)
5.1.1 Bi-Infinite Symbolic Sequences	227	(2)
5.1.2 Decomposition of the Phase Plane	229	(1)
5.1.3 Tangencies and Admissibility Conditions	230	(1)
5.1.4 Admissibility Condition in Symbolic Plane	231	(2)
5.2 Invariant Manifolds and Dynamical Foliations of Phase Plane	233	(7)
5.2.1 Stable and Unstable Invariant Manifolds	233	(3)
5.2.2 Dynamical Foliations of the Phase Plane	236	(2)
5.2.3 Summary and Discussion	238	(2)
5.3 The Tel Map	240	(16)
5.3.1 Forward and Backward Symbolic Sequences	241	(1)
5.3.2 Dynamical Foliations of Phase Space and Their Ordering	242	(7)
5.3.3 Forbidden and Allowed Zones in Symbolic Plane	249	(3)
5.3.4 The Admissibility Conditions	252	(3)
5.3.5 Summary	255	(1)
5.4 The Lozi Map	256	(19)
5.4.1 Forward and Backward Symbolic Sequences	258	(1)
5.4.2 Dynamical Foliations of the Phase Space	259	(8)
5.4.3 Ordering of the Forward and Backward Foliations	267	(2)
5.4.4 Allowed and Forbidden Zones in Symbolic Plane	269	(4)
5.4.5 Discussion of the Admissibility Condition	273	(2)
5.5 The Henon Map	275	(12)
5.5.1 Fixed Points and Their Stability	276	(2)
5.5.2 Determination of Partition Lines in Phase Plane	278	(4)
5.5.3 Henon-Type Symbolic Dynamics	282	(1)
5.5.4 Symbolic Analysis at Typical Parameter Values	283	(4)
5.6 The Dissipative Standard Map	287	(7)
5.6.1 Dynamical Foliations of the Phase Plane	287	(2)
5.6.2 Ordering of Symbolic Sequences	289	(2)
5.6.3 Symbolic Plane and Admissibility of Symbolic Sequences	291	(3)
5.7 The Stadium Billiard Problem	294	(9)
5.7.1 A Coding Based on Lifting	295	(3)
5.7.2 Relation to Other Codings	298	(2)
5.7.3 The Half-Stadium	300	(1)
5.7.4 Summary	301	(2)
6 Application to Ordinary Differential Equations	303	(52)
6.1 General Discussion	305	(7)
6.1.1 Three Types of ODEs	305	(1)
6.1.2 On Numerical Integration of Differential Equations	306	(2)
6.1.3 Numerical Calculation of the Poincare Maps	308	(4)
6.2 The Periodically Forced Brusselator	312	(14)
6.2.1 The Brusselator Viewed from The Standard Map	314	(3)
6.2.2 Transition from Annular to Interval Dynamics	317	(4)
6.2.3 Symbolic Analysis of Interval Dynamics	321	(5)
6.3 The Lorenz Equations	326	(26)
6.3.1 Summary of Known Properties	328	(2)
6.3.2 Construction of Poincare and Return Maps	330	(4)
6.3.3 One-Dimensional Symbolic Dynamics Analysis	334	(3)
6.3.4 Symbolic Dynamics of the 2D Poincare Maps	337	(8)
6.3.5 Stable Periodic Orbits	345	(7)
6.3.6 Concluding Remarks	352	(1)
6.4 Summary of Other ODE Systems	352	(3)
6.4.1 The Driven Two-Well Duffing Equation	353	(1)
6.4.2 The NMR-Laser Model	354	(1)
7 Counting the Number of Periodic Orbits	355	(30)
7.1 Periodic versus Chaotic Regimes	355	(7)
7.1.1 Stable Versus Unstable Periods in 1D Maps	356	(2)
7.1.2 Notations and Summary of Results	358	(3)
7.1.3 A Few Number Theory Notations and Functions	361	(1)
7.2 Number of Periodic Orbits in a Class of One-Parameter Maps	362	(11)
7.2.1 Number of Admissible Words in Symbolic Dynamics	362	(1)
7.2.2 Number of Tangent and Period-Doubling Bifurcations	363	(2)
7.2.3 Recursion Formula for the Total Number of Periods	365	(1)
7.2.4 Symmetry Types of Periodic Sequences	366	(4)
7.2.5 Explicit Solutions to the Recurrence Relations	370	(1)
7.2.6 Finite Lambda Auto-Expansion of Real Numbers	371	(2)
7.3 Other Aspects of the Counting Problem	373	(2)
7.3.1 The Number of Roots of the "Dark Line" Equation	373	(1)
7.3.2 Number of Saddle Nodes in Forming Smale Horseshoe	373	(1)
7.3.3 Number of Solutions of Renormalization Group Equations	374	(1)
7.4 Counting Formulae for General Continuous Maps	375	(2)
7.5 Number of Periods in Maps With Discontinuity	377	(4)
7.5.1 Number of Periods in the Gap Map	377	(2)
7.5.2 Number of Periods in the Lorenz-Like Map	379	(2)
7.6 Summary of the Counting Problem	381	(1)
7.7 Cycle Expansion for Topological Entropy	381	(4)
8 Symbolic Dynamics and Grammatical Complexity	385	(18)
8.1 Formal Languages and Their Complexity	386	(4)
8.1.1 Formal Language	386	(2)
8.1.2 Chomsky Hierarchy of Grammatical Complexity	388	(1)
8.1.3 The L-System	389	(1)
8.2 Regular Language and Finite Automaton	390	(5)
8.2.1 Finite Automaton	390	(1)
8.2.2 Regular Language	391	(1)
8.2.3 Stefan Matrix as Transfer Function for Automaton	391	(4)
8.3 Beyond Regular Languages	395	(7)
8.3.1 Feigenbaum and Generalized Feigenbaum Limiting Sets	396	(1)
8.3.2 Even and Odd Fibonacci Sequences	396	(2)
8.3.3 Odd Maximal Primitive Prefixes and Kneading Map	398	(3)
8.3.4 Even Maximal Primitive Prefixes and Distinct Excluded Blocks	401	(1)
8.4 Summary of Results	402	(1)
9 Symbolic Dynamics and Knot Theory	403	(10)
9.1 Knots and Links	404	(2)
9.2 Knots and Links from Unimodal Maps	406	(4)
9.3 Linking Numbers	410	(1)
9.4 Discussion	411	(2)
Appendix	413	(10)
A.1 Program to Generate Admissible Sequences	413	(6)
A.2 Program to Draw Dynamical Foliations of a 2D Map	419	(4)
References	423	(16)
R.1 Books	423	(1)
R.2 Papers	424	(15)
Subject Index	439