Nonlinear Dynamics: Integrability, Chaos and Patterns

ISBN-13: 9783540439080 / Angielski / Twarda / 2002 / 620 str.

L. M. Brekhovskikh; Muthusamy Lakshmanan; Shanmuganathan Rajaseekar

Opis

This self-contained treatment covers all aspects of nonlinear dynamics, from fundamentals to recent developments, in a unified and comprehensive way. Numerous examples and exercises will help the student to assimilate and apply the techniques presented.

Recenzje

From the reviews:

"The authors must be congratulated ... . The book contains many examples, exercises and problems. ... Many technical details are relegated to appendices, making the main text highly readable. ... There are a large number of nice figures and illustrations, which complement the text very well. ... I feel that this book would be a valuable addition to a personal or departmental library. ... I believe that this book has the potential to become a standard text in the ... field of non-linear dynamics." (Sanjay Puri, International Journal of Robust and Nonlinear Control, Vol. 15 (11), 2005)

"The book is an extensive treatise of nonlinear dynamical systems with emphasis on the concepts of chaos, integrability and patterns. ... the book contains numerous examples and exercises divided in two groups by their difficulty." (Peter Polacik, Zentralblatt MATH, Vol. 1038 (13), 2004)

"The book gives a comprehensive introduction to the different fields in nonlinear dynamics, such as chaos, fractals, integrability and soliton theory. It also includes a large number of interesting applications. ... Each chapter and the appendix include a number of exercises. ... the book can be highly recommended for beginners in these fields since it provides a good survey of the different fields in nonlinear dynamics." (W.-H. Steeb, Mathematical Reviews, 2004 d)

"Undoubtedly, the best survey text on Nonlinear Dynamics available today. ... the survey is carefully balanced, with detail and clarity, to enable students and researchers from other fields to learn the salient tools. For advanced undergraduate students and graduate research students looking for just one book in Nonlinear Dynamics, here is the one to get. ... It is precisely the book that has been needed to provide students with the fundamental knowledge across the landscape of Nonlinear Dynamics ... ." (B I Henry, The Physicist, Vol. 40 (4), 2003)

Spis treści

1. What is Nonlinearity?.- 1.1 Dynamical Systems: Linear and Nonlinear Forces.- 1.2 Mathematical Implications of Nonlinearity.- 1.2.1 Linear and Nonlinear Systems.- 1.2.2 Linear Superposition Principle.- 1.3 Working Definition of Nonlinearity.- 1.4 Effects of Nonlinearity.- 2. Linear and Nonlinear Oscillators.- 2.1 Linear Oscillators and Predictability.- 2.1.1 Free Oscillations.- 2.1.2 Damped Oscillations.- 2.1.3 Damped and Forced Oscillations.- 2.2 Damped and Driven Nonlinear Oscillators.- 2.2.1 Free Oscillations.- 2.2.2 Damped Oscillations.- 2.2.3 Forced Oscillations — Primary Resonance and Jump Phenomenon (Hysteresis).- 2.2.4 Secondary Resonances (Subharmonic and Superharmonic).- 2.3 Nonlinear Oscillations and Bifurcations.- Problems.- 3. Qualitative Features.- 3.1 Autonomous and Nonautonomous Systems.- 3.2 Dynamical Systems as Coupled First-Order Differential Equations: Equilibrium Points.- 3.3 Phase Space/Phase Plane and Phase Trajectories: Stability, Attractors and Repellers.- 3.4 Classification of Equilibrium Points: Two-Dimensional Case.- 3.4.1 General Criteria for Stability.- 3.4.2 Classification of Equilibrium (Singular) Points.- 3.5 Limit Cycle Motion — Periodic Attractor.- 3.5.1 Poincaré-Bendixson Theorem.- 3.6 Higher Dimensional Systems.- 3.6.1 Example: Lorenz Equations.- 3.7 More Complicated Attractors.- 3.7.1 Torus.- 3.7.2 Quasiperiodic Attractor.- 3.7.3 Poincaré Map.- 3.7.4 Chaotic Attractor.- 3.8 Dissipative and Conservative Systems.- 3.8.1 Hamiltonian Systems.- 3.9 Conclusions.- Problems.- 4. Bifurcations and Onset of Chaos in Dissipative Systems.- 4.1 Some Simple Bifurcations.- 4.1.1 Saddle-Node Bifurcation.- 4.1.2 The Pitchfork Bifurcation.- 4.1.3 Transcritical Bifurcation.- 4.1.4 Hopf Bifurcation.- 4.2 Discrete Dynamical Systems.- 4.2.1 The Logistic Map.- 4.2.2 Equilibrium Points and Their Stability.- 4.2.3 Stability When the First Derivative Equals to +1 or –1.- 4.2.4 Periodic Solutions or Cycles.- 4.2.5 Period Doubling Phenomenon.- 4.2.6 Onset of Chaos: Sensitive Dependence on Initial Conditions — Lyapunov Exponent.- 4.2.7 Bifurcation Diagram.- 4.2.8 Bifurcation Structure in the Interval 3.57 ? a ? 4.- 4.2.9 Exact Solution at a = 4.- 4.2.10 Logistic Map: A Geometric Construction of the Dynamics — Cobweb Diagrams.- 4.3 Strange Attractor in the H´enon Map.- 4.3.1 The Period Doubling Phenomenon.- 4.3.2 Self-Similar Structure.- 4.4 Other Routes to Chaos.- 4.4.1 Quasiperiodic Route to Chaos.- 4.4.2 Intermittency Route to Chaos.- 4.4.3 Type-I Intermittency.- 4.4.4 Standard Bifurcations in Maps.- Problems.- 5. Chaos in Dissipative Nonlinear Oscillators and Criteria for Chaos.- 5.1 Bifurcation Scenario in Duffing Oscillator.- 5.1.1 Period Doubling Route to Chaos.- 5.1.2 Intermittency Transition.- 5.1.3 Quasiperiodic Route to Chaos.- 5.1.4 Strange Nonchaotic Attractors (SNAs).- 5.2 Lorenz Equations.- 5.2.1 Period Doubling Bifurcations and Chaos.- 5.3 Some Other Ubiquitous Chaotic Oscillators.- 5.3.1 Driven van der Pol Oscillator.- 5.3.2 Damped, Driven Pendulum.- 5.3.3 Morse Oscillator.- 5.3.4 Rössler Equations.- 5.4 Necessary Conditions for Occurrence of Chaos.- 5.4.1 Continuous Time Dynamical Systems (Differential Equations).- 5.4.2 Discrete Time Systems (Maps).- 5.5 Computational Chaos, Shadowing and All That.- 5.6 Conclusions.- Problems.- 6. Chaos in Nonlinear Electronic Circuits.- 6.1 Linear and Nonlinear Circuit Elements.- 6.2 Linear Circuits: The Resonant RLC Circuit.- 6.3 Nonlinear Circuits.- 6.3.1 Chua’s Diode: Autonomous Case.- 6.3.2 A Simple Practical Implementation of Chua’s Diode.- 6.3.3 Bifurcations and Chaos.- 6.4 Chaotic Dynamics of the Simplest Dissipative Nonautonomous Circuit: Murali-Lakshmanan-Chua (MLC) Circuit.- 6.4.1 Experimental Realization.- 6.4.2 Stability Analysis.- 6.4.3 Explicit Analytical Solutions.- 6.4.4 Experimental and Numerical Studies.- 6.5 Analog Circuit Simulations.- 6.6 Some Other Useful Nonlinear Circuits.- 6.6.1 RL Diode Circuit.- 6.6.2 Hunt’s Nonlinear Oscillator.- 6.6.3 p-n Junction Diode Oscillator.- 6.6.4 Modified Chua Circuit.- 6.6.5 Colpitt’s Oscillator.- 6.7 Nonlinear Circuits as Dynamical Systems.- Problems.- 7. Chaos in Conservative Systems.- 7.1 Poincaré Cross Section or Surface of Section.- 7.2 Possible Orbits in Conservative Systems.- 7.2.1 Regular Trajectories.- 7.2.2 Irregular Trajectories.- 7.2.3 Canonical Perturbation Theory: Overlapping Resonances and Chaos.- 7.3 Hénon-Heiles System.- 7.3.1 Equilibrium Points.- 7.3.2 Poincaré Surface of Section of the System.- 7.3.3 Numerical Results.- 7.4 Periodically Driven Undamped Duffing Oscillator.- 7.5 The Standard Map.- 7.5.1 Linear Stability and Invariant Curves.- 7.5.2 Numerical Analysis: Regular and Chaotic Motions.- 7.6 Kolmogorov-Arnold-Moser Theorem.- 7.7 Conclusions.- Problems.- 8. Characterization of Regular and Chaotic Motions.- 8.1 Lyapunov Exponents.- 8.2 Numerical Computation of Lyapunov Exponents.- 8.2.1 One-Dimensional Map.- 8.2.2 Computation of Lyapunov Exponents for Continuous Time Dynamical Systems.- 8.3 Power Spectrum.- 8.3.1 The Power Spectrum and Dynamical Motion.- 8.4 Autocorrelation.- 8.5 Dimension.- 8.6 Criteria for Chaotic Motion.- Problems.- 9. Further Developments in Chaotic Dynamics.- 9.1 Time Series Analysis.- 9.1.1 Estimation of Time-Delay and Embedding Dimension.- 9.1.2 Largest Lyapunov Exponent.- Problems.- 9.2 Stochastic Resonance.- Problems.- 9.3 Chaotic Scattering.- Problems.- 9.4 Controlling of Chaos.- 9.4.1 Controlling and Controlling Algorithms.- 9.4.2 Stabilization of UPO.- Problems.- 9.5 Synchronization of Chaos.- 9.5.1 Chaos in the DVP Oscillator.- 9.5.2 Synchronization of Chaos in the DVP Oscillator.- 9.5.3 Chaotic Signal Masking and Transmission of Analog Signals.- 9.5.4 Chaotic Digital Signal Transmission.- Problems.- 9.6 Quantum Chaos.- 9.6.1 Quantum Signatures of Chaos.- 9.6.2 Rydberg Atoms and Quantum Chaos.- 9.6.3 Hydrogen Atom in a Generalized van der Waals Interaction.- 9.6.4 Outlook.- Problems.- 9.7 Conclusions.- 10. Finite Dimensional Integrable Nonlinear Dynamical Systems.- 10.1 What is Integrability?.- 10.2 The Notion of Integrability.- 10.3 Complete Integrability — Complex Analytic Integrability.- 10.3.1 Real Time and Complex Time Behaviours.- 10.3.2 Partial Integrability and Constrained Integrability.- 10.3.3 Integrability and Separability.- 10.4 How to Detect Integrability: Painlevé Analysis.- 10.4.1 Classification of Singular Points.- 10.4.2 Historical Development of the Painlevé Approach and Integrability of Ordinary Differential Equations.- 10.4.3 Painlevé Method of Singular Point Analysis for Ordinary Differential Equations.- 10.5 Painlevé Analysis and Integrability of Two-Coupled Nonlinear Oscillators.- 10.5.1 Quartic Anharmonic Oscillators.- 10.6 Symmetries and Integrability.- 10.6.1 Invariance Conditions, Determination of Infinitesimals and First Integrals of Motion.- 10.6.2 Application — The Hénon-Heiles System.- 10.7 A Direct Method of Finding Integrals of Motion.- 10.8 Integrable Systems with Degrees of Freedom Greater Than Two.- 10.9 Integrable Discrete Systems.- 10.10 Integrable Dynamical Systems on Discrete Lattices.- 10.11 Conclusion.- Problems.- 11. Linear and Nonlinear Dispersive Waves.- 11.1 Linear Waves.- 11.2 Linear Nondispersive Wave Propagation.- 11.3 Linear Dispersive Wave Propagation.- 11.4 Fourier Transform and Solution of Initial Value Problem.- 11.5 Wave Packet and Dispersion.- 11.6 Nonlinear Dispersive Systems.- 11.6.1 An Illustration of the Wave of Permanence.- 11.6.2 John Scott Russel’s Great Wave of Translation.- 11.7 Cnoidal and Solitary Waves.- 11.7.1 Korteweg—de Vries Equation and the Solitary Waves and Cnoidal Waves.- 11.8 Conclusions.- Problems.- 12. Korteweg—de Vries Equation and Solitons.- 12.1 The Scott Russel Phenomenon and KdV Equation.- 12.2 The Fermi-Pasta-Ulam Numerical Experiments on Anharmonic Lattices.- 12.2.1 The FPU Lattice.- 12.2.2 FPU Recurrence Phenomenon.- 12.3 The KdV Equation Again!.- 12.3.1 Asymptotic Analysis and the KdV Equation.- 12.4 Numerical Experiments of Zabusky and Kruskal: The Birth of Solitons.- 12.5 Hirota’s Direct or Bilinearization Method for Soliton Solutions of KdV Equation.- 12.6 Conclusions.- 13. Basic Soliton Theory of KdV Equation.- 13.1 The Miura Transformation and Linearization of KdV: The Lax Pair.- 13.1.1 The Miura Transformation.- 13.1.2 Galilean Invariance and Schrödinger Eigenvalue Problem.- 13.1.3 Linearization of the KdV Equation.- 13.1.4 Lax Pair.- 13.2 Lax Pair and the Method of Inverse Scattering: A New Method to Solve the Initial Value Problem.- 13.2.1 The Inverse Scattering Transform (IST) Method for KdV Equation.- 13.3 Explicit Soliton Solutions.- 13.3.1 One-Soliton Solution (N = 1).- 13.3.2 Two-Soliton Solution.- 13.3.3 N-Soliton Solution.- 13.3.4 Soliton Interaction.- 13.3.5 Nonreflectionless Potentials.- 13.4 Hamiltonian Structure of KdV Equation.- 13.4.1 Dynamics of Continuous Systems.- 13.4.2 KdV as a Hamiltonian Dynamical System.- 13.4.3 Complete Integrability of the KdV Equation.- 13.5 Infinite Number of Conserved Densities.- 13.6 Bäcklund Transformations.- 13.7 Conclusions.- 14. Other Ubiquitous Soliton Equations.- 14.1 Identification of Some Ubiquitous Nonlinear Evolution Equations from Physical Problems.- 14.1.1 The Nonlinear Schrödinger Equation in Optical Fibers.- 14.1.2 The Sine-Gordon Equation in Long Josephson Junctions.- 14.1.3 Dynamics of Ferromagnets: Heisenberg Spin Equations.- 14.1.4 The Lattice with Exponential Interaction: The Toda Equation.- 14.2 The Zakharov-Shabat (ZS)/ Ablowitz-Kaup-Newell-Segur (AKNS) Linear Eigenvalue Problem and NLEES.- 14.2.1 The AKNS Linear Eigenvalue Problem and AKNS Equations.- 14.2.2 The Standard Soliton Equations.- 14.3 Solitary Wave Solutions and Basic Solitons.- 14.3.1 The MKdV Equation: Pulse Soliton.- 14.3.2 The sine-Gordon Equation: Kink, Antikink and Breathers.- 14.3.3 The Nonlinear Schr¨odinger Equation: Envelope Soliton.- 14.3.4 The Heisenberg Spin Equation: The Spin Soliton.- 14.3.5 The Toda Lattice: Discrete Soliton.- 14.4 Hirota’s Method and Soliton Nature of Solitary Waves.- 14.4.1 The Modified KdV Equation.- 14.4.2 The NLS Equation.- 14.4.3 The sine-Gordon Equation.- 14.4.4 The Heisenberg Spin System.- 14.5 Solutions via IST Method.- 14.5.1 Direct and Inverse Scattering.- 14.5.2 Time Evolution of the Scattering Data.- 14.5.3 Soliton Solutions.- 14.6 Bäcklund Transformations.- 14.7 Conservation Laws and Constants of Motion.- 14.8 Hamiltonian Structure and Integrability.- 14.8.1 Hamiltonian Structure.- 14.8.2 Complete Integrability of the NLS Equation.- 14.9 Conclusions.- Problems.- 15. Spatio-Temporal Patterns.- 15.1 Linear Diffusion Equation.- 15.2 Nonlinear Diffusion and Reaction-Diffusion Equations.- 15.2.1 Nonlinear Reaction-Diffusion Equations.- 15.2.2 Dissipative Systems.- 15.3 Spatio-Temporal Patterns in Reaction-Diffusion Systems.- 15.3.1 Homogeneous Patterns.- 15.3.2 Autowaves: Travelling Wave Fronts, Pulses, etc.- 15.3.3 Ring Waves, Spiral Waves and Scroll Waves.- 15.3.4 Turing Instability and Turing Patterns.- 15.3.5 Localized Structures.- 15.3.6 Spatio-Temporal Chaos.- 15.4 Cellular Neural/Nonlinear Networks (CNNs).- 15.4.1 Cellular Nonlinear Networks (CNNs).- 15.4.2 Arrays of MLC Circuits: Simple Examples of CNN.- 15.4.3 Active Wave Propagation and its Failure in One-Dimensional CNNs.- 15.4.4 Turing Patterns.- 15.4.5 Spatio-Temporal Chaos.- 15.5 Some Exactly Solvable Nonlinear Diffusion Equations.- 15.5.1 The Burgers Equation.- 15.5.2 The Fokas-Yortsos-Rosen Equation.- 15.5.3 Generalized Fisher’s Equation.- 15.6 Conclusion.- Problems.- 16. Nonlinear Dynamics: From Theory to Technology.- 16.1 Chaotic Cryptography.- 16.1.1 Basic Idea of Cryptography.- 16.1.2 An Elementary Chaotic Cryptographic System.- 16.2 Using Chaos (Controlling) to Calm the Web.- 16.3 Some Other Possibilities of Using Chaos.- 16.3.1 Communicating by Chaos.- 16.3.2 Chaos and Financial Markets.- 16.4 Optical Soliton Based Communications.- 16.5 Soliton Based Optical Computing.- 16.5.1 Photo-Refractive Materials and the Manakov Equation.- 16.5.2 Soliton Solutions and Shape Changing Collisions.- 16.5.3 Optical Soliton Based Computation.- 16.6 Micromagnetics and Magnetoelectronics.- 16.7 Conclusions.- A. Elliptic Functions and Solutions of Certain Nonlinear Equations.- Problems.- B. Perturbation and Related Approximation Methods.- B.1 Approximation Methods for Nonlinear Differential Equations.- B.2 Canonical Perturbation Theory for Conservative Systems.- B.2.1 One Degree ol Freedom Hamiltonian Systems.- B.2.2 Two Degrees ol Freedom Systems.- Problems.- C. A Fourth-Order Runge-Kutta Integration Method.- Problems.- Problems.- E. Fractals and Multifractals.- Problems.- Problems.- G. Inverse Scattering Transform for the Schrödinger Spectral Problem.- G.l The Linear Eigenvalue Problem.- G.2 The Direct Scattering Problem.- G.3 The Inverse Scattering Problem.- G.4 Reconstruction of the Potential.- Problems.- H. Inverse Scattering Transform for the Zakharov-Shabat Eigenvalue Problem.- H.1 The Linear Eigenvalue Problem.- H.2 The Direct Scattering Problem.- H.3 Inverse Scattering Problem.- H.4 Reconstruction of the Potentials.- Problems.- I. Integrable Discrete Soliton Systems.- I.1 Integrable Finite Dimensional N-Particles System on a Line: Calogero-Moser System.- I.2 The Toda Lattice.- I.3 Other Discrete Lattice Systems.- I.4 Solitary Wave (Soliton) Solution of the Toda Lattice.- Problems.- J. Painlevé Analysis for Partial Differential Equations.- J.1 The Painlevé Property for PDEs.- J.1.1 Painlevé Analysis.- J.2 Examples.- J.2.1 KdV Equation.- J.2.2 The Nonlinear Schrödinger Equation.- Problems.- References.

Blurb

Integrability, chaos and patterns are three of the most important concepts in nonlinear dynamics. These are covered in this book from fundamentals to recent developments. The book presents a self-contained treatment of the subject to suit the needs of students, teachers and researchers in physics, mathematics, engineering and applied sciences who wish to gain a broad knowledge of nonlinear dynamics. It describes fundamental concepts, theoretical procedures, experimental and numerical techniques and technological applications of nonlinear dynamics. Numerous examples and problems are included to facilitate the understanding of the concepts and procedures described. In addition to 16 chapters of main material, the book contains 10 appendices which present in-depth mathematical formulations involved in the analysis of various nonlinear systems.