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Devaney R.L. — An introduction to chaotic dynamical systems
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Название: An introduction to chaotic dynamical systemsАвтор: Devaney R.L. Аннотация: The study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book. In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets. This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry, Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas. The first two chapters introduce the reader to a broad spectrum of fundamental topics in dynamics: hyperbolicity, symbolic dynamics, structural stability, stable and unstable manifolds and bifurcation theory. Readers familiar with linear algebra and complex analysis will be led to the brink of contemporary research in the books concluding chapter, but for anyone with a background in calculus, Devaney provides a comprehensive exploration into the mathematics of chaos.
Язык: Рубрика: Физика /Нелинейная динамика, Хаос /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: second editionГод издания: 1989Количество страниц: 336Добавлена в каталог: 16.08.2005Операции: Положить на полку |
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Предметный указатель
8 54 Adding machine 137 Admissible sequence 144 Analytic map 261 Anosov map 190 Area preserving map 255 Asymptotic 19 185 Attracting fixed point 25 215 276 Attractor 25 201 Backward asymptotic 19 185 Baker map 39 52 Basin of attraction 280 Bifurcation 28 80 Bifurcation diagram 82 Bifurcation, Homoclinic 126 Bifurcation, period-doubling 82 90 310 Bifurcation, Saddle node 81 88 310 Block, L. 63 Branched manifold 209 Brouwer fixed point theorem 14 Bump function 8 17 Canonical family 109 Cantor bouquet 326 Cantor function 111 Cantor Middle-Fifths set 39 Cantor Middle-Thirds set 36 138 Cantor set 37 187 270 Cartesian product 171 Chain recurrence 232 Chain rule 10 Chaos 50 268 Closed set 14 Closing Lemma 116 Completely invariant 269 composition 10 Cone 238 Cone field 238 Continued fraction 307 Continuous dynamical system 3 Contraction mapping theorem 172 Core 238 Cover 61 Covering map 102 Critical point 10 DA map 212 Degenerate critical point 19 Degenerate homoclinic orbit 124 Dendrite 294 Denjoy map 108 112 Dense orbit 42 Dense set 15 Devil’s staircase 110 Diffeomorphism 9 170 Discrepancy 142 Discrete dynamical system 2 Douady, A. 260 316 double 67 137 Dynamical system 2 Eigenvalue 164 Eigenvector 164 Elliptic point 255 Elliptic transformation 267 Escape rate 314 Eventually periodic point 18 Exceptional point 274 Expanding attractor 208 Expansive 50 External ray 312 Fatou, P. 260 Fibonacci sequence 101 Filled in Julia set 311 First return map 75 Fixed point 18 Foliation 191 Forward asymptotic 19 185 Fractal 37 Full family 153 Fundamental domain 55 genealogy 154 Gradient like 231 Graph transform 226 Graphical analysis 20 32 Guckenheimer, J. 63 Hartman’s theorem 58 Henon attractor 211 213 Henon map 170 251 Heteroclinic point 122 188 233 Homeomorphism 9 Homoclinic bifurcation 125 257 Homoclinic orbit 123 188 Homoclinic point 122 194 233 Hopf bifurcation 242—244 Hopf bifurcation theorem 249 Horizontal curve 224 Horizontal line field 228 Hubbard, J. 260 316 Hyperbolic 24—28 214 Hyperbolic, fixed point 24 215 Hyperbolic, periodic point 24 21S Hyperbolic, set 38 187 236 Hyperbolic, total automorphism 191 Hyperbolic, transformation 267 Implicit function theorem 10 171 Indifferent periodic point 276 Injective 9 integral 181 Intermediate Value Theorem 11 Inverse function theorem 172 Inverse limit 206 Involution 188 255 Irrational rotation 21 Iteration 2 Itinerary 44 Jacobi’s theorem 21 Jordan form 169 Julia set 269 Julia, G. 260 Kneading sequence 141 Kneading theory 140—143 Koenigs 276 Kupka — Smale theorem 117 Liapounov function 176 Lift 102 Limit point 14 Linear automorphism 230 Linear map 161 Linear structural stability 59 Local stable manifold 217 Local stable set 26 Local unstable manifold 217 Local unstable set 122 Lozi attractor 214 Mandelbrot set 295 299 311—317 Mandelbrot, B. 260 Mapping 17 Markov, partition 196 Matrix 161 Matrix representation 163 Maximum principle 263 Mean value theorem 10 Metric 40 Minimal set 136 Misiurewicz, M. 63 137 320 Mobius transformation 267 Montel’s theorem 274 Morse — Smale Map 59 114 235 Moser Twist Theorem 257 Moser, J. 276 Neutral periodic point 300 Non-degenerate critical point 19 Non-degenerate homoclinic orbit 124 Non-wandering 47 Normal family 272 Normal form 245 One-to-one 8 Onto 9 Open set 15 Orbit 17 Orbit diagram 134 Orbit, backward orbit 17 Orbit, forward orbit 17 Orbit, recurrent orbit 47 115 Orientation preserving 102 Palis, J. 235 Parabolic transformation 267 Perfect set 37 Period-doubling bifurcation 82 90 130 240 310 Periodic point 18 Periodic point, attracting 25 Periodic point, indifferent 276 300 Periodic point, neutral 300 Periodic point, repelling 26 Periodic point, weakly attracting 28 Periodic point, weakly repelling 27 Petal 302 Phase portrait 20 Plykin attractor 209 Pole 266 Quadratic map 31—39 268—272 Rational rotation 21 Recurrent point 47 116 232 Regular sequence 138 Renormalization 133 146 Repelling periodic point 26 215 Repellor 26 Reversible 255 Riemann sphere 265 Rotation number 103 Saddle node bifurcation 80 88 240 310 Sarkovskii order 62 256 Sarkovskii’s theorem 60 99 Schroder functional equation 277 Schwarz lemma 264 Schwarzian derivative 68—79 268 Sector bundle 223 Semi-conjugate 51 Sensitive dependence 49 Sequence space 40 184 Shift map 40 184 270 Siegel, C.L. 276 306 Simply connected 262 Sink 25 215 Smale horseshoe map 180 Smale, S. 93 Smooth function 8 Snap-back repellor 122 Solenoid 201 Source 26 215 Stable manifold 218 237 Stable set 19 185 269 Stable subspace 177 Standard family 109 Standard form 167 Steiner circle 268 Strange Attractor 211 258 Structural stability 53 Subshift of finite type 94 199 Sullivan, D. 260 Superattracting 276 Surjective 9 Symbolic dynamics 39—42 184 256 Tangent bifurcation 81 Tchebycheff polynomial 52 Tent map 38 52 Topological conjugacy 47 Topological transitivity 42 49 Torus 171 190 Totally disconnected 37 Trace 99 Transition family 153 Transition matrix 94 Transitive attractor 204 Transversality 233 Transverse homoclinic point 194 Trapping region 202 Uniformizing map 314 Unimodal map 130 140 Unstable manifold 218 237 Unstable set 19 186 Unstable subspace 177 Wandering interval 80 109 Weierstrass p function 296 Williams, R. 93 208 Young, L.-S. 63
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