Chaos and nonlinear dynamics : an introduction for scientists and engineers / Robert C. Hilborn.

By: Hilborn, Robert C .

Material type: materialTypeLabel

BookPublisher: New York ; Oxford : Oxford University Press, 1994Description: xvii, 654 p. : ill ; 25 cm. + hbk.ISBN: 0195057600; 0195088166 .Subject(s): Chaotic behavior in systems

| Quantum chaos

| Dynamics

| Nonlinear theories

DDC classification: 003.75

Contents:

I. The Phenomenology of Chaos -- Three chaotic systems -- The universality of chaos -- II. Toward a theory of nonlinear dynamics and chaos -- Dynamics in state space: One and Two dimensions -- Three-dimensional state space and chaos -- Iterated maps -- Quasi-periodicity and chaos -- Intermittency and Crises -- Hamiltonian systems -- III. Measures of Chaos -- Quantifying chaos -- Many dimensions and multifractals -- IV. Special Topics -- Pattern Formation and Spatiotemporal chaos -- Quantum chaos, the theory of complexity and other topics.

Holdings
Item type	Current library	Call number	Copy number	Status	Date due	Barcode	Item holds
General Lending	MTU Bishopstown Library Lending	003.75 (Browse shelf(Opens below))	1	Available		00017868
General Lending	MTU Bishopstown Library Store Item	003.75 (Browse shelf(Opens below))	1	Available		00017869

Total holds: 0

Enhanced descriptions from Syndetics:

This is the only book that introduces the full range of activity in the rapidly growing field of nonlinear dynamics to an audience of students, scientists, and engineers with no in-depth experience in the area. The text uses a step-by-step explanation of dynamics and geometry in state space as a foundation for understanding nonlinear dynamics. It goes on to provide a thorough treatment of such key topics as differential equation models and iterated map models (including a derivation of the famous Feigenbaum numbers), the surprising role of number theory in dynamics, and an introduction to Hamiltonian dynamics. This is the only book written at this introductory level to include the increasingly important field of pattern formation, along with a survey of the controversial questions of quantum chaos. Important analytical tools, such as Lyapunov exponents, Kolmogorov entropies, and fractal dimensions, are treated in detail. With over 200 figures and diagrams, and both analytic and computer exercises following every chapter, the book is ideally suited for use as a text or for self-instruction. An extensive collection of annotated references brings the reader into contact with the literature in nonlinear dynamics, which the reader will be prepared to tackle after completing the book.

Includes bibliographical references (pages 628-648) and index.

Table of contents provided by Syndetics

First Edition Preface (p. v)
First Edition Acknowledgments (p. xi)
Second Edition Preface (p. xiii)
Second Edition Acknowledgments (p. xv)
I. The Phenomenology of Chaos (p. 1)
1 Three Chaotic Systems (p. 3)
1.1 Prelude (p. 3)
1.2 Linear and Nonlinear Systems (p. 4)
1.3 A Nonlinear Electrical System (p. 8)
1.4 A Mathematical Model of Biological Population Growth (p. 17)
1.5 A Model of Convecting Fluids: The Lorenz Model (p. 27)
1.6 Determinism, Unpredictability, and Divergence of Trajectories (p. 37)
1.7 Summary and Conclusions (p. 39)
1.8 Further Reading (p. 40)
2 The Universality of Chaos (p. 47)
2.1 Introduction (p. 47)
2.2 The Feigenbaum Numbers (p. 47)
2.3 Convergence Ratio for Real Systems (p. 51)
2.4 Using [delta] to Make Predictions (p. 53)
2.5 Feigenbaum Size Scaling (p. 55)
2.6 Self-Similarity (p. 56)
2.7 Other Universal Features (p. 57)
2.8 Models and the Universality of Chaos (p. 58)
2.9 Computers and Chaos (p. 61)
2.10 Further Reading (p. 63)
2.11 Computer Exercises (p. 64)
II. Toward a Theory of Nonlinear Dynamics and Chaos (p. 69)
3 Dynamics in State Space: One and Two Dimensions (p. 71)
3.1 Introduction (p. 71)
3.2 State Space (p. 72)
3.3 Systems Described by First-Order Differential Equations (p. 74)
3.4 The No-Intersection Theorem (p. 77)
3.5 Dissipative Systems and Attractors (p. 78)
3.6 One-Dimensional State Space (p. 79)
3.7 Taylor Series Linearization Near Fixed Points (p. 83)
3.8 Trajectories in a One-Dimensional State Space (p. 84)
3.9 Dissipation Revisited (p. 86)
3.10 Two-Dimensional State Space (p. 87)
3.11 Two-Dimensional State Space: The General Case (p. 91)
3.12 Dynamics and Complex Characteristic Values (p. 94)
3.13 Dissipation and the Divergence Theorem (p. 96)
3.14 The Jacobian Matrix for Characteristic Values (p. 97)
3.15 Limit Cycles (p. 100)
3.16 Poincare Sections and the Stability of Limit Cycles (p. 102)
3.17 Bifurcation Theory (p. 106)
3.18 Summary (p. 113)
3.19 Further Reading (p. 114)
3.20 Computer Exercises (p. 116)
4 Three-Dimensional State Space and Chaos (p. 117)
4.1 Overview (p. 117)
4.2 Heuristics (p. 118)
4.3 Routes to Chaos (p. 121)
4.4 Three-Dimensional Dynamical Systems (p. 123)
4.5 Fixed Points in Three Dimensions (p. 124)
4.6 Limit Cycles and Poincare Sections (p. 128)
4.7 Quasi-Periodic Behavior (p. 134)
4.8 The Routes to Chaos I: Period-Doubling (p. 136)
4.9 The Routes to Chaos II: Quasi-Periodicity (p. 137)
4.10 The Routes to Chaos III: Intermittency and Crises (p. 138)
4.11 The Routes to Chaos IV: Chaotic Transients and Homoclinic Orbits (p. 138)
4.12 Homoclinic Tangles and Horseshoes (p. 146)
4.13 Lyapunov Exponents and Chaos (p. 148)
4.14 Further Reading (p. 154)
4.15 Computer Exercises (p. 155)
5 Iterated Maps (p. 157)
5.1 Introduction (p. 157)
5.2 Poincare Sections and Iterated Maps (p. 158)
5.3 One-Dimensional Iterated Maps (p. 163)
5.4 Bifurcations in Iterated Maps: Period-Doubling, Chaos, and Lyapunov Exponents (p. 166)
5.5 Qualitative Universal Behavior: The U-Sequence (p. 173)
5.6 Feigenbaum Universality (p. 183)
5.7 Tent Map (p. 185)
5.8 Shift Maps and Symbolic Dynamics (p. 188)
5.9 The Gaussian Map (p. 192)
5.10 Two-Dimensional Iterated Maps (p. 197)
5.11 The Smale Horseshoe Map (p. 199)
5.12 Summary (p. 204)
5.13 Further Reading (p. 204)
5.14 Computer Exercises (p. 207)
6 Quasi-Periodicity and Chaos (p. 210)
6.1 Introduction (p. 210)
6.2 Quasi-Periodicity and Poincare Sections (p. 212)
6.3 Quasi-Periodic Route to Chaos (p. 214)
6.4 Universality in the Quasi-Periodic Route to Chaos (p. 215)
6.5 Frequency-Locking (p. 217)
6.6 Winding Numbers (p. 218)
6.7 Circle Map (p. 219)
6.8 The Devil's Staircase and the Farey Tree (p. 227)
6.9 Continued Fractions and Fibonacci Numbers (p. 231)
6.10 On to Chaos and Universality (p. 234)
6.11 Some Applications (p. 240)
6.12 Further Reading (p. 246)
6.13 Computer Exercises (p. 249)
7 Intermittency and Crises (p. 250)
7.1 Introduction (p. 250)
7.2 What Is Intermittency? (p. 250)
7.3 The Cause of Intermittency (p. 252)
7.4 Quantitative Theory of Intermittency (p. 256)
7.5 Types of Intermittency and Experimental Observations (p. 259)
7.6 Crises (p. 260)
7.7 Some Conclusions (p. 267)
7.8 Further Reading (p. 268)
7.9 Computer Exercises (p. 270)
8 Hamiltonian Systems (p. 272)
8.1 Introduction (p. 272)
8.2 Hamilton's Equations and the Hamiltonian (p. 273)
8.3 Phase Space (p. 276)
8.4 Constants of the Motion and Integrable Hamiltonians (p. 279)
8.5 Nonintegrable Systems, the KAM Theorem, and Period-Doubling (p. 289)
8.6 The Henon-Heiles Hamiltonian (p. 296)
8.7 The Chirikov Standard Map (p. 303)
8.8 The Arnold Cat Map (p. 308)
8.9 The Dissipative Standard Map (p. 309)
8.10 Applications of Hamiltonian Dynamics (p. 311)
8.11 Further Reading (p. 313)
8.12 Computer Exercises (p. 316)
III. Measures of Chaos (p. 317)
9 Quantifying Chaos (p. 319)
9.1 Introduction (p. 319)
9.2 Time-Series of Dynamical Variables (p. 320)
9.3 Lyapunov Exponents (p. 323)
9.4 Universal Scaling of the Lyapunov Exponent (p. 327)
9.5 Invariant Measure (p. 330)
9.6 Kolmogorov-Sinai Entropy (p. 335)
9.7 Fractal Dimension(s) (p. 341)
9.8 Correlation Dimension and a Computational Case History (p. 354)
9.9 Comments and Conclusions (p. 368)
9.10 Further Reading (p. 369)
9.11 Computer Exercises (p. 374)
10 Many Dimensions and Multifractals (p. 375)
10.1 General Comments and Introduction (p. 375)
10.2 Embedding (Reconstruction) Spaces (p. 376)
10.3 Practical Considerations for Embedding Calculations (p. 383)
10.4 Generalized Dimensions and Generalized Correlation Sums (p. 389)
10.5 Multifractals and the Spectrum of Scaling Indices f([alpha]) (p. 393)
10.6 Generalized Entropy and the g([lambda]) Spectrum (p. 404)
10.7 Characterizing Chaos via Periodic Orbits (p. 413)
10.8 Statistical Mechanical and Thermodynamic Formalism (p. 415)
10.9 Wavelet Analysis, q-Calculus, and Related Topics (p. 420)
10.10 Summary (p. 421)
10.11 Further Reading (p. 422)
10.12 Computer Exercises (p. 429)
IV. Special Topics (p. 431)
11 Pattern Formation and Spatiotemporal Chaos (p. 433)
11.1 Introduction (p. 433)
11.2 Two-Dimensional Fluid Flow (p. 436)
11.3 Coupled-Oscillator Models, Cellular Automata, and Networks (p. 442)
11.4 Transport Models (p. 450)
11.5 Reaction-Diffusion Systems: A Paradigm for Pattern Formation (p. 460)
11.6 Diffusion-Limited Aggregation, Dielectric Breakdown, and Viscous Fingering: Fractals Revisited (p. 471)
11.7 Self-Organized Criticality: The Physics of Fractals? (p. 477)
11.8 Summary (p. 479)
11.9 Further Reading (p. 480)
11.10 Computer Exercises (p. 489)
12 Quantum Chaos, The Theory of Complexity, and Other Topics (p. 490)
12.1 Introduction (p. 490)
12.2 Quantum Mechanics and Chaos (p. 490)
12.3 Chaos and Algorithmic Complexity (p. 508)
12.4 Miscellaneous Topics: Piece-wise Linear Models, Time-Delay Models, Information Theory, Stochastic Resonance, Computer Networks, Controlling and Synchronizing Chaos (p. 510)
12.5 Roll Your Own: Some Simple Chaos Experiments (p. 517)
12.6 General Comments and Overview: The Future of Chaos (p. 517)
12.7 Further Reading (p. 519)
Appendix A Fourier Power Spectra (p. 533)
Appendix B Bifurcation Theory (p. 541)
Appendix C The Lorenz Model (p. 547)
Appendix D The Research Literature on Chaos (p. 559)
Appendix E Computer Programs (p. 560)
Appendix F Theory of the Universal Feigenbaum Numbers (p. 568)
Appendix G The Duffing Double-Well Oscillator (p. 579)
Appendix H Other Universal Features for One-Dimensional Iterated Maps (p. 584)
Appendix I The van der Pol Oscillator (p. 589)
Appendix J Simple Laser Dynamics Models (p. 598)
References (p. 605)
Index (p. 643)

Reviews provided by Syndetics

CHOICE Review

Hilborn's book is superb. Written by a master teacher and researcher, it is a model of thorough exposition. In addition to an excellent bibliography, every chapter contains an annotated list of references. In Chapter 2, there is a list of five inexpensive software packages for nonlinear dynamics and chaos calculations. In succeeding chapters, computer exercises using this software form a vital part of the strategy for the study of this fascinating field. Each chapter has, in addition, exercises that range from verifying a result just stated to studies that extend and exemplify the text. The painstaking development of new ideas relies heavily on previous sections and is facilitated by graphs, figures, and sketches wherever appropriate. The presentation assumes a familiarity with both introductory college physics and calculus up through elementary differential equations. Readers with such a background will find this book a treasure for both self-study and classroom use. At present, this book has no peers. Upper-division undergraduate through faculty. C. A. Hewett; Rochester Institute of Technology

Author notes provided by Syndetics

Robert C. Hilborn is Amanda and Lisa Cross Professor of Physics at Amherst College