Nonlinear Dynamics: Integrability, Chaos and Patterns (Hardcover, 2003)

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 Poincare-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 Poincare 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 Rossler 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