Book:D.W. Jordan/Nonlinear Ordinary Differential Equations/Second Edition

D.W. Jordan and P. Smith: Nonlinear Ordinary Differential Equations, 2nd ed.

Published $\text {1987}$, Oxford University Press

ISBN 0-19-859656-1

Subject Matter

• Non-Linear Differential Equations

Contents

Preface

1 SECOND-ORDER DIFFERENTIAL EQUATIONS IN THE PHASE PLANE

1.1. Phase diagram for the pendulum equation

1.2. Autonomous equations in the phase plane

1.3. Conservative systems

1.4. The damped linear oscillator

1.5. Nonlinear damping

1.6. Some applications

1.7. Parameter-dependent conservative systems

Exercises

2. FIRST-ORDER SYSTEMS IN TWO VARIABLES AND LINEARIZATION

2.1. The general phase plane

2.2. Some population models

2.3. Linear approximation at equilibrium points

2.4. The general solution of a linear system

2.5. Classifying equilibrium points

2.6. Constructing a phase diagram

2.7. Transitions between types of equilibrium point

Exercises

3. GEOMETRICAL AND COMPUTATIONAL ASPECTS OF THE PHASE DIAGRAM

3.1. The index of a point

3.2. The index at infinity

3.3. The phase diagram at infinity

3.4. Llmit cycles and other closed paths

3.5. Computation of the phase diagram

Exercises

4. AVERAGING METHODS

4.1. An energy-balance method for limit cycles

4.2. Amplitude and frequency estimates

4.3. Slowly-varying amplitude : nearly periodic solutions

4.4. Penodic solutions: harmonic balance

4.5. The equivalent linear equation by harmonic balance

Exercises

5. PERTURBATION METHODS

5.1. Outline of the direct method

5.2. Forced oscillations far from resonance

5.3. Forced oscillations near resonance with weak excitation

5.4. The amplitude equation for the undamped pendulum

5.5. The amplitude equation for a damped pendulum

5.6. Soft and hard springs

5.7. Amplitude-phase perturbation for the pendulum equation

5.8. Periodic solutions of autonomous equations (Lindstedt's method)

5.9. Forced oscillation of a self-excited equation

5.10. The perturbation method and Fourier series

Exercises

6. SINGULAR PERTURBATION M ETHODS

6.1. Non-uniform approximations to functions on an interval

6.2. Coordinate perturbation (renormalization)

6.3. Lighthill's method

6.4. Multiple time scales (two-timing)

6.5. Matching approximations on an interval

6.6. A matching technique for differential equations

Exercises

7. FORCED OSCILLATIONS: HARMONIC AND SUBHARMONIC RESPONSE. STABILITY. ENTRAINMENT

7.1. General forced periodic solutions

7.2. Harmonic solutions, transients, and stability for Duffing's equation

7.3. The jump phenomenon

7.4. Harmonic oscillations, stability, and transients for the forced van der Pol equation

7.5. Frequency entrainment for the van der Pol equation

7.6. Comparison of the theory with computations

7.7. Subharmonics of Duffing's equation by perturbation

7.8. Stability and transients for subharmonics of Duffing's equation

Exercises

8. STABILITY

8.1. Poincaré stability (stability of paths)

8 2. Paths and solution curves

8.3. Liapunov stability (stability of solutions)

8.4. Stability of linear systems

8.5. Structure of the solutions of $n$-dimensional linear systems

8.6. Stability and boundedness for linear systems

8.7. Stability of systems with constant coefficients

Exercises

9. DETERMINATION OF STABILITY BY SOLUTION PERTURBATION

9.1. The stability of forced oscillations by solution perturbation

9.2. Equations with periodic coefficients (Floquet theory)

9.3. Mathieu's equation arising from a Duffing equation

9.4. Transition curves for Mathieu's equation by perturbation

9.5. Mathieu's damped equation arising from a Duffing equation

Exercises

10. LIAPUNOV METHODS FOR DETERMINING STABILITY

10.1. Liapunov's direct method

10.2. Liapunov functions

10.3. A test for instability

10.4. Stability and the linear approximation in two dimensions

10.5. Special systems

Exercises

11. THE EXISTENCE OF PERIODIC SOLUTIONS

11.1. The Poincaré-Bendixson theorem

11.2. A theorem on the existence of a centre

11.3. A theorem on the existence of a limit cycle

11.4. Van der Pol's equation with large parameter

Exercises

12. BIFURCATIONS, STRUCTURAL STABILITY, AND CHAOS

12.1. Examples of bifurcations

12.2. The fold and the cusp

12.3. Structural stability and bifurcations

12.4. Hopf bifurcations

12.5. Poincaré maps

12.6. Chaos and strange attractors

12.7. Perturbation analysis of an amplitude bifurcation

12.8. Homoclinic bifurcation

Exercises

APPENDIX A: Existence and uniqueness theorems

APPENDIX B: Hints and answers to the exercises

BIBLIOGRAPHY

INDEX