## An Introduction to Dynamical SystemsIn recent years there has been an explosion of research centred on the appearance of so-called 'chaotic behaviour'. This book provides a largely self contained introduction to the mathematical structures underlying models of systems whose state changes with time, and which therefore may exhibit this sort of behaviour. The early part of this book is based on lectures given at the University of London and covers the background to dynamical systems, the fundamental properties of such systems, the local bifurcation theory of flows and diffeomorphisms, Anosov automorphism, the horseshoe diffeomorphism and the logistic map and area preserving planar maps . The authors then go on to consider current research in this field such as the perturbation of area-preserving maps of the plane and the cylinder. This book, which has a great number of worked examples and exercises, many with hints, and over 200 figures, will be a valuable first textbook to both senior undergraduates and postgraduate students in mathematics, physics, engineering, and other areas in which the notions of qualitative dynamics are employed. |

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### Contents

II | 1 |

III | 5 |

IV | 6 |

V | 11 |

VI | 16 |

VII | 20 |

VIII | 28 |

IX | 33 |

XLI | 206 |

XLIII | 211 |

XLIV | 215 |

XLV | 218 |

XLVI | 221 |

XLVII | 226 |

XLVIII | 234 |

XLIX | 245 |

X | 38 |

XI | 42 |

XII | 56 |

XIII | 64 |

XV | 67 |

XVI | 69 |

XVII | 72 |

XVIII | 79 |

XIX | 83 |

XX | 89 |

XXI | 93 |

XXII | 102 |

XXIII | 105 |

XXIV | 108 |

XXV | 119 |

XXVI | 120 |

XXVII | 123 |

XXVIII | 125 |

XXIX | 132 |

XXX | 138 |

XXXI | 139 |

XXXII | 147 |

XXXIII | 149 |

XXXIV | 154 |

XXXV | 170 |

XXXVI | 180 |

XXXVII | 190 |

XXXIX | 199 |

XL | 203 |

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### Common terms and phrases

area-preserving map attracting set attractor behaviour bifurcation curve bifurcation diagram bifurcations occurring Cantor set centre manifold closed orbit complex conjugacy Consider corresponding defined Definition dense differential equation DX(x dynamics eigenvalues example exists flow follows function given Hamiltonian Hence Henon homeomorphism homoclinic points homoclinic tangle Hopf bifurcation hyperbolic fixed point illustrated implies integer intersect invariant circle invariant set irrational rotation island chains isoclines iterations Jordan form lift limit cycle linear diffeomorphism linearisation neighbourhood non-trivial fixed points Observe obtain origin parameter periodic points phase portrait planar Poincare map Proposition resonant terms result rotation interval rotation number saddle connection saddle point saddle-node bifurcation satisfies sequence Show shown in Figure shown see Exercise solution stable and unstable structurally stable subset sufficiently small symplectic takes the form tangent Theorem topological type topologically conjugate topologically equivalent trajectories transformation transverse twist map unstable manifolds values vector field approximation Verify versal unfolding zero