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Springer Berlin Heidelberg
Nonlinear Dynamics: Integrability, Chaos and Patterns / Edition 1

Nonlinear Dynamics: Integrability, Chaos and Patterns / Edition 1


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Product Details

ISBN-13: 9783540439080
Publisher: Springer Berlin Heidelberg
Publication date: 12/16/2002
Series: Advanced Texts in Physics
Edition description: 2003
Pages: 620
Product dimensions: 6.10(w) x 9.25(h) x 0.06(d)

Table of Contents

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 &Hacute;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.

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