| Introduction | p. 1 |
| Solving differential equations | p. 6 |
| Existence and uniqueness theorems | p. 12 |
| Phase space and flows | p. 13 |
| Limit sets and trajectories | p. 19 |
| Exercises 1 | p. 23 |
| Stability | p. 25 |
| Definitions of stability | p. 27 |
| Liapounov functions | p. 32 |
| Strong linear stability | p. 40 |
| Orbital stability | p. 46 |
| Bounding functions | p. 47 |
| Non-autonomous equations | p. 50 |
| Exercises 2 | p. 51 |
| Linear Differential Equations | p. 54 |
| Autonomous linear differential equations | p. 55 |
| Normal forms | p. 57 |
| Invariant manifolds | p. 66 |
| Geometry of phase space | p. 69 |
| Floquet Theory | p. 70 |
| Exercises 3 | p. 73 |
| Linearization and Hyperbolicity | p. 77 |
| Poincare's Linearization Theorem | p. 78 |
| Hyperbolic stationary points and the stable manifold theorem | p. 83 |
| Persistence of hyperbolic stationary points | p. 89 |
| Structural stability | p. 90 |
| Nonlinear sinks | p. 92 |
| The proof of the stable manifold theorem | p. 96 |
| Exercises 4 | p. 100 |
| Two Dimensional Dynamics | p. 102 |
| Linear systems in R[superscript 2] | p. 102 |
| The effect of nonlinear terms | p. 109 |
| Rabbits and sheep | p. 119 |
| Trivial linearization | p. 124 |
| The Poincare index | p. 126 |
| Dulac's criterion | p. 129 |
| Pike and eels | p. 130 |
| The Poincare-Bendixson Theorem | p. 132 |
| Decay of large amplitudes | p. 137 |
| Exercises 5 | p. 141 |
| Periodic Orbits | p. 145 |
| Linear and nonlinear maps | p. 146 |
| Return maps | p. 150 |
| Floquet Theory revisited | p. 151 |
| Periodically forced differential equations | p. 156 |
| Normal forms for maps in R[superscript 2] | p. 157 |
| Exercises 6 | p. 158 |
| Perturbation Theory | p. 161 |
| Asymptotic expansions | p. 162 |
| The method of multiple scales | p. 165 |
| Multiple scales for forced oscillators: complex notation | p. 170 |
| Higher order terms | p. 172 |
| Interlude | p. 172 |
| Nearly Hamiltonian systems | p. 176 |
| Resonance in the Mathieu equation | p. 179 |
| Parametric excitation in Mathieu equations | p. 180 |
| Frequency locking | p. 181 |
| Hysteresis | p. 184 |
| Asynchronous quenching | p. 188 |
| Subharmonic resonance | p. 190 |
| Relaxation oscillators | p. 194 |
| Exercises 7 | p. 196 |
| Bifurcation Theory I: Stationary Points | p. 199 |
| Centre manifolds | p. 201 |
| Local bifurcations | p. 206 |
| The saddlenode bifurcation | p. 208 |
| The transcritical bifurcation | p. 213 |
| The pitchfork bifurcation | p. 215 |
| An example | p. 219 |
| The Implicit Function Theorem | p. 222 |
| The Hopf bifurcation | p. 224 |
| Calculation of the stability coefficient, ReA(0) | p. 238 |
| A canard | p. 244 |
| Exercises 8 | p. 246 |
| Bifurcation Theory II: Periodic Orbits and Maps | p. 249 |
| A simple eigenvalue of +1 | p. 250 |
| Period-doubling bifurcations | p. 255 |
| The Hopf bifurcation for maps | p. 259 |
| Arnol'd (resonant) tongues | p. 267 |
| Exercises 9 | p. 271 |
| Bifurcational Miscellany | p. 274 |
| Unfolding degenerate singularities | p. 274 |
| Imperfection theory | p. 276 |
| Isolas | p. 278 |
| Periodic orbits in Lotka-Volterra models | p. 279 |
| Subharmonic resonance revisited | p. 285 |
| Chaos | p. 291 |
| Characterizing chaos | p. 293 |
| Period three implies chaos | p. 300 |
| Unimodal maps I: an overview | p. 302 |
| Tent maps | p. 307 |
| Unimodal maps II: the non-chaotic case | p. 313 |
| Quantitative universality and scaling | p. 317 |
| Intermittency | p. 324 |
| Partitions, graphs and Sharkovskii's Theorem | p. 327 |
| Exercises 11 | p. 335 |
| Global Bifurcation Theory | p. 338 |
| An example | p. 339 |
| Homoclinic orbits and the saddle index | p. 344 |
| Planar homoclinic bifurcations | p. 346 |
| Homoclinic bifurcations to a saddle-focus | p. 351 |
| Lorenz-like equations | p. 359 |
| Cascades of homoclinic bifurcations | p. 367 |
| Exercises 12 | p. 373 |
| Notes and further reading | p. 377 |
| Bibliography | p. 382 |
| Index | p. 386 |
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