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

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## D.W. Jordan and P. Smith:

## 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

### 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. PoincarĂ© 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 PoincarĂ©-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. PoincarĂ© 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