Oscillations in Planar Dynamic Systems

The following sections are included:

• PREFACE

• TABLE OF CONTENTS

The following sections are included:

• Introduction

• Examples of Nonlinear Systems
  • The Pendulum
  • Mass Attached to a Stretched Wire
  • Vibrations of the Eardrum
  • Nonlinear Electrical Circuits
  • Negative-Resistance Oscillator
  • Oscillations of a Diatomic Molecule
  • Nonlinear Oscillators with Damping
  • Brusselator Model
  • Glycolysis
  • Chlorine Dioxide-Iodine Reaction
  • Advertising Model
  • Predator-Prey

• Dimensionless Form of Differential Equations
  • Linear Damped Oscillator
  • Nonlinear Oscillator
  • Rayleigh Equation

• Nonlinear Equations Having Exact Solutions
  • Harmonic Oscillator
  • Particle-in-a-Box
  • Antisymmetric, Constant Force Oscillator
  • d²y/dt² + y + ∊y³ = 0
  • Exact Period of the Pendulum
  • Pendulum

• Problems

• References

The following sections are included:

• Introduction

• Secular Terms

• Lindstedt-Poincaré Method

• Examples
  • Example A
  • Example B
  • Example C
  • Example D
  • Example E
  • Discussion

• Existence of a Periodic Solution
  • Two Conditions
  • F a Function Only of y
  • F = F₁(y) - C(dy/dt)
  • F a Function Only of dy/dt
  • F = F₃(y)dy/dt

• Shohat Expansion

• Problems

• References

The following sections are included:

• Introduction

• First Approximation of Krylov and Bogoliubov
  • Technique
  • Two Special Cases
  • Stability Properties of Limit Cycles
  • Equivalent Linearization

• Worked Examples Using the Method of Krylov and Bogoliubov
  • Example A
  • Example B
  • Example C
  • Example D
  • Example E
  • Example F
  • Example G
  • Example H

• Method of Krylov-Bogoliubov-Mitropolsky

• Worked Examples Using the Method of Krylov, Bogoliubov, and Mitropolsky
  • Example A
  • Example B
  • Example C
  • Example D

• Spurious Limit Cycles

• Problems

• References

The following sections are included:

• Introduction

• General Method
  • Bounds on the Fourier Coefficients
  • Direct Harmonic Balance
  • Rational Representations

• Worked Examples: First Approximation
  • Example A
  • Example B
  • Example C
  • Example D
  • Example E
  • Example F
  • Example G
  • Example H

• Comparison with the Lindstedt-Poincaré Method

• Worked Examples: Second Approximation
  • Example A
  • Example B

• Problems

• References

The following sections are included:

• Introduction

• Two-Time Expansion

• Worked Examples Using the Two-Time Expansion
  • Example A
  • Example B
  • Example C
  • Example D
  • Example E
  • Example F

• Derivative Expansion Procedure

• Worked Examples Using the Derivative Expansion Procedure
  • Example A
  • Example B
  • Example C

• Problems

• References

The following sections are included:

• Introduction

• Glycolytic Oscillator

• Hopf Bifurcation Theorem
  • The Theorem
  • Example

• General Procedure for Two-Variable Systems

• The Recipe

• Worked Examples
  • van der Pol Equation
  • WCM Oscillator
  • Batch Fermentation
  • Brusselator Model

• Problems

• References

The following sections are included:

• Appendix A MATHEMATICAL RELATIONS
  • Trigonometric Relations
    • Exponential Definitions of Tirgonometric Functions
    • Functions of Sums of Angles
    • Powers of Trigonometric Functions
    • Other Trigonometric Relations
    • Derivatives and Integrals of Trigonometric Functions
  • Factors and Expansions
  • Quadratic Equations
  • Cubic Equations
  • Differentiation of a Definite Integral with Respect to a Parameter
  • Eigenvalues of a 2 × 2 Matrix
  • References

• Appendix B SERIES EXPANSIONS
  • Uniform Convergence
  • Weierstrass M Test for Uniform Convergence
  • Properties of Uniformly Convergent Series
  • Power Series
  • Taylor Series of a Function of a Single Variable
  • Generalized Taylor's Theorem
  • Taylor Series of a Function of Two Variables
  • References

• Appendix C FOURIER SERIES
  • Definition of Fourier Series
  • Convergence of Fourier Series
    • Examples
    • Convergence Theorem
  • Bounds on Fourier Coefficients
  • Expansion of F(a cos x, -a sin x) in a Fourier Series
  • References

• Appendix D ASYMPTOTICS EXPANSIONS
  • Gauge Functions and Order Symbols
    • The Symbol O
    • The Symbols o
  • Asymptotic Expansion
  • Uniform Expansions
  • Elementary Operations on Asymptotic Expansions
    • Addition and Subtraction
    • Integration
    • Multiplication
    • Differentiation
  • Examples
    • The Symbol O
    • The Symbol o
    • The Function sin(x + ∊)
    • The Function exp(-∊x)
  • Generalized Asymptotic Power Series
  • References

• Appendix E BASIC THEOREMS OF THE THEORY OF SECOND-ORDER DIFFERENTIAL EQUATIONS
  • Introduction
  • Existence and Uniqueness of the Solution
  • Dependence of the Solution on Initial Conditions
  • Dependence of the Solution on a Parameter
  • References

• Appendix F LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS
  • Basic Existence Theorem
  • Homogeneous Linear Differential Equations
    • Linear Combination
    • Linear Dependent and Linear Independent Functions
    • Theorems on Linear Second-Order Homogeneous Differential Equations
  • Inhomogeneous Linear Differential Equations
    • Principle of Superposition
    • Solutions of Linear Inhomogeneous Differential Equations
  • Linear Second-Order Homogeneous Differential Equations with Constant Coefficients
  • Linear Second-Order Inhomogeneous Differential Equations with Constant Coefficients
  • References

• Appendix G EXISTENCE OF PERIODIC SOLUTIONS FOR CERTAIN SECOND-ORDER DIFFERENTIAL EQUATIONS
  • Limit Cycles
  • Liénard-Levinson-Smith Theorem
  • Levinson-Smith Theorem
  • References

• Appendix H STABILITY OF LIMIT CYCLES
  • Introduction
  • Stability Condition
  • References

• Appendix I QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS
  • Introduction
  • Basic Procedure
  • Applications
  • References

• GENERAL BIBLIOGRAPHY

• INDEX