Nonlinear Dynamics

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.

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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.

Contenu

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 …