Nonlinear System Analysis

Foreword

Preface

Chapter 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

References

Chapter 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

References

Chapter 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

References

Chapter 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

References

Chapter 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

References

Chapter 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

References

Chapter 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

References

Chapter 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

References

Chapter 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

References

Appendix. 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

References

Author Index

Subject Index