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

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