Nonlinear System Analysis

ForewordPrefaceChapter I. Linearity and Nonlinearity 1. An Example of a Nonlinear System: The Simple Pendulum 2. Conservative Oscillators 3. Approximate Solutions of the Pendulum Equation 4. Exact Solution by Elliptic Integral 5. Representation in a Phase Plane 6. Nonlinear Oscillator with Damping 7. Simple Pendulum with Forcing Function. Resonance ReferencesChapter II. Self-Oscillatory Systems Introduction 1. Electronic Oscillators 2. Phase-Plane Representation 3. Cauchy-Lipschitz Theorem 4. Geometric Study of Periodic Solutions 5. Analytic Approaches to Periodic Phenomena 6. Synchronization of Self-Oscillators 7. Subharmonic Response ReferencesChapter III. Classification of Singularities 1. Singular Points 2. Distribution of Singular Points in Phase-Plane R2 3. Static and Dynamic Systems 4. Extension of the Theory: Sources, Sinks, and Transformation Points 5. Transformations of the Vector Field 6. Three-Dimensional Singularities ReferencesChapter IV. Systems with Several Degrees of Freedom 1. Introduction 2. Example of a Conservative Oscillator 3. Nonlinear Oscillations in a Particle Accelerator 4. Self-Sustained Oscillators with Two Degrees of Freedom 5. Normal Vibrations on Nonlinear Systems ReferencesChapter V. Equivalent Linearization 1. Stating the Problem 2. A Model in Classical Optics 3. Introduction to the Optimal Linearization Method 4. Similarity with Fourier's Method 5. Optimal Linear Operator 6. Iteration of the Procedure 7. The Describing Function 8. Additive Property of the Describing Function 9. Matrix Calculus in the Analysis of Nonlinear Systems ReferencesChapter VI. The Describing Function Method 1. Equation of Feedback Loops 2. Linear and Nonlinear Feedback Loops 3. Nyquist's Diagram 4. Mikaïlov's Hodograph 5. Generalization of Mikaïlov's Hodograph for Nonlinear Systems 6. Applications to Autonomous Systems 7. Applications to Nonautonomous Systems 8. Sensitivity with Respect to Small Changes in Parameters 9. Retarded Actions 10. Multiple-Input Describing Function ReferencesChapter VII. Nonlinear Equations with Periodic Coefficients Introduction 1. Perturbation Method 2. Stepwise Method: Application to the Orbital Stability Problem in a Synchrotron 3. Hamiltonian Representation 4. The Smooth Approximation ReferencesChapter VIII. System Response to Random Inputs 1. Campbell's Theorem 2. Fokker-Planck-Kolmogorov Method 3. Solution of the Fokker-Planck-Kolmogorov Equation Based on Campbell's Theorem ReferencesChapter IX. Random Fluctuations of Self-Oscillators Introduction 1. Berstein's Method 2. Blaquière's Method 3. Lerner's Quasi-Linear Method 4. Flicker Noise 5. Error in Frequency Measurement Using a Finite Time t' 6. Application to Masers ReferencesAppendix. Sinusoidal Modes of Electromagnetic Resonators 1. Equation for Linear Oscillations 2. Nonlinear Oscillations: Single Mode 3. Synchronization of Two Modes, Spatially Separated, in the Nonlinear Region 4. Synchronization of Two Modes, Nonspatially Separated, in the Nonlinear Region; Coupling by the Nonlinearity Only ReferencesAuthor IndexSubject Index