# 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.

### 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