| Preface | |
| Diffeomorphisms and flows | p. 1 |
| Introduction | p. 1 |
| Elementary dynamics of diffeomorphisms | p. 5 |
| Definitions | p. 5 |
| Diffeomorphisms of the circle | p. 6 |
| Flows and differential equations | p. 11 |
| Invariant sets | p. 16 |
| Conjugacy | p. 20 |
| Equivalence of flows | p. 28 |
| Poincare maps and suspensions | p. 33 |
| Periodic non-autonomous systems | p. 38 |
| Hamiltonian flows and Poincare maps | p. 42 |
| Exercises | p. 56 |
| Local properties of flows and diffeomorphisms | p. 64 |
| Hyperbolic linear diffeomorphisms and flows | p. 64 |
| Hyperbolic non-linear fixed points | p. 67 |
| Diffeomorphisms | p. 68 |
| Flows | p. 69 |
| Normal forms for vector fields | p. 72 |
| Non-hyperbolic singular points of vector fields | p. 79 |
| Normal forms for diffeomorphisms | p. 83 |
| Time-dependent normal forms | p. 89 |
| Centre manifolds | p. 93 |
| Blowing-up techniques on R[superscript 2] | p. 102 |
| Polar blowing-up | p. 102 |
| Directional blowing-up | p. 105 |
| Exercises | p. 108 |
| Structural stability, hyperbolicity and homoclinic points | p. 119 |
| Structural stability of linear systems | p. 120 |
| Local structural stability | p. 123 |
| Flows on two-dimensional manifolds | p. 125 |
| Anosov diffeomorphisms | p. 132 |
| Horseshoe diffeomorphisms | p. 138 |
| The canonical example | p. 139 |
| Dynamics on symbol sequences | p. 147 |
| Symbolic dynamics for the horseshoe diffeomorphism | p. 149 |
| Hyperbolic structure and basic sets | p. 154 |
| Homoclinic points | p. 164 |
| The Melnikov function | p. 170 |
| Exercises | p. 180 |
| Local bifurcations I: planar vector fields and diffeomorphisms on R | p. 190 |
| Introduction | p. 190 |
| Saddle-node and Hopf bifurcations | p. 199 |
| Saddle-node bifurcation | p. 199 |
| Hopf bifurcation | p. 203 |
| Cusp and generalised Hopf bifurcations | p. 206 |
| Cusp bifurcation | p. 206 |
| Generalised Hopf bifurcations | p. 211 |
| Diffeomorphisms on R | p. 215 |
| D[subscript x]f(0) = +1: the fold bifurcation | p. 218 |
| D[subscript x]f(0) = -1: the flip bifurcation | p. 221 |
| The logistic map | p. 226 |
| Exercises | p. 234 |
| Local bifurcations II: diffeomorphisms on R[superscript 2] | p. 245 |
| Introduction | p. 245 |
| Arnold's circle map | p. 248 |
| Irrational rotations | p. 253 |
| Rational rotations and weak resonance | p. 258 |
| Vector field approximations | p. 262 |
| Irrational [beta] | p. 262 |
| Rational [beta] = p/q, q [greater than or equal] 3 | p. 264 |
| Rational [beta] = p/q, q = 1, 2 | p. 268 |
| Equivariant versal unfoldings for vector field approximations | p. 271 |
| q = 2 | p. 272 |
| q = 3 | p. 275 |
| q = 4 | p. 276 |
| q [greater than or equal] 5 | p. 282 |
| Unfoldings of rotations and shears | p. 286 |
| Exercises | p. 291 |
| Area-preserving maps and their perturbations | p. 302 |
| Introduction | p. 302 |
| Rational rotation numbers and Birkhoff periodic points | p. 309 |
| The Poincare-Birkhoff Theorem | p. 309 |
| Vector field approximations and island chains | p. 310 |
| Irrational rotation numbers and the KAM Theorem | p. 319 |
| The Aubry-Mather Theorem | p. 332 |
| Invariant Cantor sets for homeomorphisms on S[superscript 1] | p. 332 |
| Twist homeomorphisms and Mather sets | p. 335 |
| Generic elliptic points | p. 338 |
| Weakly dissipative systems and Birkhoff attractors | p. 345 |
| Birkhoff periodic orbits and Hopf bifurcations | p. 355 |
| Double invariant circle bifurcations in planar maps | p. 368 |
| Exercises | p. 379 |
| Hints for exercises | p. 394 |
| References | p. 413 |
| Index | p. 417 |
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