An Introduction to Dynamical SystemsCambridge University Press, 27 juil. 1990 - 423 pages In 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. |
Table des matières
II | 1 |
III | 5 |
IV | 6 |
V | 11 |
VI | 16 |
VII | 20 |
VIII | 28 |
IX | 33 |
XL | 206 |
XLII | 211 |
XLIII | 215 |
XLIV | 218 |
XLV | 221 |
XLVI | 226 |
XLVII | 234 |
XLVIII | 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 |
XXXVIII | 199 |
XXXIX | 203 |
XLIX | 248 |
L | 253 |
LI | 258 |
LII | 262 |
LIV | 264 |
LV | 268 |
LVI | 271 |
LVII | 272 |
LVIII | 275 |
LIX | 276 |
LX | 282 |
LXI | 286 |
LXII | 291 |
LXIII | 302 |
LXV | 309 |
LXVI | 310 |
LXVII | 319 |
LXVIII | 332 |
LXX | 335 |
LXXI | 338 |
LXXII | 345 |
LXXIII | 355 |
LXXIV | 368 |
LXXV | 379 |
LXXVI | 394 |
LXXVII | 413 |
417 | |
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Expressions et termes fréquents
a₁ area-preserving area-preserving map behaviour bifurcation curve bifurcation diagram Birkhoff periodic Cantor set centre manifold complex Consider coordinates corresponding defined differential equation dynamics eigenvalues example flow follows function given Hamiltonian Hence homeomorphism homoclinic points homoclinic tangle Hopf bifurcation hyperbolic fixed point implies intersect invariant circle irrational rotation island chains isoclines iterations limit cycle m₁ m₂ neighbourhood neZ+ non-hyperbolic non-trivial fixed points normal form obtain P₁ parameter period q periodic orbit periodic points phase portrait planar Poincaré map polar Proposition rational resonant terms rotation number saddle connection saddle point saddle-node satisfies sequence Show shown in Figure singularity solution stable and unstable structurally stable subset sufficiently small Theorem topological type topologically conjugate topologically equivalent trajectories transformation transverse twist map unstable manifolds v₁ v₂ vector field approximation Verify versal unfolding x₁ y₁ zero