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Hilborn R.C. — Chaos and nonlinear dynamics
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Íàçâàíèå: Chaos and nonlinear dynamicsÀâòîð: Hilborn R.C. Àííîòàöèÿ: This is the only book that introduces the full range of activity in the rapidly growing field of nonlinear dynamics to an audience of students, scientists, and engineers with no in-depth experience in the area. The text uses a step-by-step explanation of dynamics and geometry in state space as a foundation for understanding nonlinear dynamics. It goes on to provide a thorough treatment of such key topics as differential equation models and iterated map models (including a derivation of the famous Feigenbaum numbers), the surprising role of number theory in dynamics, and an introduction to Hamiltonian dynamics. This is the only book written at this introductory level to include the increasingly important field of pattern formation, along with a survey of the controversial questions of quantum chaos. Important analytical tools, such as Lyapunov exponents, Kolmogorov entropies, and fractal dimensions, are treated in detail. With over 200 figures and diagrams, and both analytic and computer exercises following every chapter, the book is ideally suited for use as a text or for self-instruction. An extensive collection of annotated references brings the reader into contact with the literature in nonlinear dynamics, which the reader will be prepared to tackle after completing the book.
ßçûê: Ðóáðèêà: Ôèçèêà /Íåëèíåéíàÿ äèíàìèêà, Õàîñ /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Èçäàíèå: 2nd editionÃîä èçäàíèÿ: 2000Êîëè÷åñòâî ñòðàíèö: 650Äîáàâëåíà â êàòàëîã: 10.09.2005Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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393—404 404—412 1/f noise 256 479 action 280—285 491 505—506 Action-angle variables 280—289 algorithmic complexity 508—510 Antimonotonicity 197 Area preserving maps 303—309 Arnold cat map 308—309 499 Arnold diffusion 294 Arnold tongues 225 231 372 Asymptotically stable 164 Attracting set see "Attractor(s)" Attractor(s) 22 32 67 78—79 Attractor(s), chaotic, defined 172 Attractor(s), strange, defined 342 Autocorrelation function 383—384 Autocorrelation time 383—388 Autonomous system 76 Average Lyapunov exponent 172 324 419 Averaging method 592—596 Baker's transformation 208 Band merging 194 Bare winding number 221 Basin boundaries 79 352—353 Basin of attraction 22 32 79 107 Benard See "Rayleigh — Benard" Bernoulli shift 190—192 Bifurcation 166—171 Bifurcation diagram 11 15—18 24—25 107—108 Bifurcation diagram, diode circuit 17—18 Bifurcation diagram, Gaussian map 194—196 Bifurcation diagram, Henon map 200 Bifurcation diagram, logistic map 24—25 48 58 59 180 263 Bifurcation diagram, sine-circle map 237—238 Bifurcation theory 106—113 541—546 Bifurcation, defined 11—12 Bifurcation, global 122 541 545 Bifurcation, Hopf 111—113 137 Bifurcation, local 121—122 Bifurcation, pitchfork see "Period-doubling" Bifurcation, point 106—113 Bifurcation, saddle-node 109 Bifurcation, tangent 109 254—256 259 Billiards 311 Biological models and noise 243—244 Birkhoff series 281 Bouncing ball model 517 Boundary crisis 260—262 Bounded system 85 Boussinesq approximation 550 Box-Counting Dimension 342—344 390 Brusselator model 100 109—110 Butterfly effect 38 Canonical transformation 280—282 Cantor set 344—345 Cantor set, asymmetric and weighted 398—402 Capacity dimension see "Boxcounting dimension" Cardiac cells 241—242 Cat map 308—309 Cellular automata 445—448 Center (fixed point) see "Elliptic point" Center manifold 543 Chaos 3—9 Chaos, criteria for 150 172 Chaos, definition 6—7 Chaotic attractor 120 Chaotic bands 25 179—182 Chaotic scattering 479 Chaotic transients 67 118 122 145 Characteristic direction 88 91 99 Characteristic equation 93 97—98 Characteristic exponent 105—106 Characteristic multiplier 105 113 132 Characteristic value 80—81 84 94 99 Chemical reactions 91 Chirikov standard map 304—307 Circle map 219—227 Cluster of initial conditions 86—87 96 150—152 159—160 Codimension 541 Coexistence of regular and chaotic motion 289—292 299—303 Coherent structures 435 Commensurate frequencies 211 Comparison time 384 Complex eigenvalues 94—96 complexity 39 434—435 490 Composition rule 169 573 Computer networks 514—515 Computer programs 64—66 560—567 Computers and chaos 61—63 Conservative system 272—273 Constant energy surface 276 Continued fractions 231—234 Continuity equation 452 Control parameter 11 15 30 Controlling chaos 515—516 Convection see "Lorenz model" Convergent 232 Correlation dimensions see "Dimensions correlation" Correlation integral 355 Correlation sum 355 (see also "Generalized correlation sum") Correspondence principle 493 Coupled modes 465 Coupled oscillator models 442—445 Crisis 122 138 260—268 Crisis-induced intermittency 265 Critical point 173 Critical state 478 criticality 240 Cycles see "Limit cycles" Damping see "Dissipation" Darcy's law 475 Degrees of freedom 72—74 274 276 Delay-differential equation 490 Dendrites 473 Determinant 98—100 Determinism 3 6 37—39 Deterministic chaos 6 Devil's staircase 227 Diffusion 450—456 Diffusion-limited aggregation 471—474 Dimensions, box-counting 342 390 Dimensions, correlation 354—368 392—394 Dimensions, fractal 341—354 Dimensions, generalized 389 390 392—394 Dimensions, Hausdorff 354 383 Dimensions, information 392 409 Dimensions, Lyapunov 383 Dimensions, pointwise 359 Dimensions, similarity 348—352 Dimensions, topological 354 Diode circuit 8—17 51—52 160—161 510—511 dispersion relation 465 Dissipation see also "Dissipative systems" Dissipation and Jacobian 97—99 Dissipation and sum of Lyapunov exponents 148—152 Dissipative standard map 309—311 Dissipative systems 78—79 86—87 Divergence of nearby trajectories 13—14 16 25 37—39 172 Divergence theorem and dissipation 87 96—97 Double crisis 267 Doubling transformation 574 Drift ring 214 227 Duffing oscillator 71 579—583 Dynamical localization 507 Dynamical partition function 416 Dynamical spectrum 409—412 Dynamical system 74 Dynamical systems theory 74 Eigenvalues see "Characteristic values" Eigenvectors 99 Electrodeposition 474—477 Elliptic point 283 305—306 438—439 Embedding method 375—389 Embedding space 375—389 Energy eigenvalues 415 502—503 Entropy, definition 356 Entropy, generalized 393—404 Entropy, information 409 Entropy, Kolmogorov — Sinai 335—341 404 419 Entropy, topological 406 409 414 Equilibrium point see "Fixed point" Ergodic, ergodicity 288 333—334 Eulerian viewpoint 278 453 Existence and uniqueness theorem 77 Exponential divergence see "Divergence of nearby trajectories" False nearest neighbors 381 Farey tree 229—231 Fat fractal 346 Feigenbaum numbers, 55—56 183—185 568—574 Feigenbaum numbers, 47—51 574—578 Feigenbaum universality 568—578 Feigenbaum, M. 17 47—48 Feigenvalues 577 Feynman 433 518 Fibonacci numbers 235 Fick's law of diffusion 451 Filtering data 365—366 Fixed point 20—22 32 164 Fixed point in three dimensions 124—128 Fixed point in two dimensions 88—94 97—99 Fixed point, elliptic 438—439 Fixed point, hyperbolic 91 148 439—440 Flip bifurcations see "Period-doubling" Floquet matrix 131 Floquet multipliers see "Characteristic multipliers" Fluid flow 436—441 Focus see "Node" Fold bifurcation see "Bifurcations saddle-node" Forced van der Pol oscillator 244 Fourier analysis and synthesis 533—540 Fractal 34 57 79 Fractal basin boundary 79 Fractal dimensions 341—354 Fractal, definition 342 Frequency-locking 135 217 Frequency-ratio parameter 218 221 Galerkin truncation 554 Game of life 447—448 Gaussian map 192—197 Generalized correlation sum 389 393 408 Generalized dimensions see "Dimensions generalized" Generalized entropies see "Entropy generalized" Gibbs 71 Ginzburg — Landau equation 469 Glass — Mackey model 512 Global bifurcation 122 541 545 Global phase portrait 85 Golden mean 231—234 Granular flow 441—442 Hamilton's equations 274 Hamiltonian function 274—276 437—438 494—496 Hamiltonian system 272—313 harmonic oscillator see "Oscillator harmonic" Hausdorff dimension 354 383 Heartbeat experiment 241—242 Heaviside step function 356 Heisenberg uncertainty relation 497 Hele-Shaw cell 475 Henon map 198 Henon — Heiles model 296—303 Heteroclinic connection 142 Heteroclinic intersection 140 Heteroclinic orbit 138—146 Heteroclinic point 140 Heteroclinic tangle 141—142 146—148 440 Homoclinic connection 141 Homoclinic intersection 140 Homoclinic orbit 138—146 Homoclinic point 140 Homoclinic tangle 141—142 146—147 440 Hopf bifurcation 111—113 137 Hopf bifurcation and intermittency 259 Horseshoe map 199—204 Horseshoe, connection to chaotic behavior 146—148 Husimi distribution 501 Hydrodynamic derivative 278 Hyperbolic fixed point 91 148 Hyperbolic point (for Hamiltonian systems) 287 439—440 Hyperchaos 150 Hysteresis 199 580 Images of the critical point 181 In-set 90—91 Incommensurate frequencies 211 Index of a fixed point 126 Information and chaos 513—514 Information, definition 513 Information, dimension 392 409 Information, entropy 409 Initial condition see "Divergence of nearby trajectories" Integrable system 273 279—289 Interior crisis 260—267 Intermittency 250—267 Intermittency, route 122 138 250—260 Intermittency, Type I, II, III 122 250—260 Invariant distribution see "Invariant measure" Invariant manifold 90 Invariant measure 330—335 Invariant measure for Bernoulli shift map 334—335 Invariant measure for logistic map 334—335 Invariant tori 288 Invertible map 198 Irrational number 188—189 211 Irrational ratio of frequencies 211 Islands in Hamiltonian systems 302 305—306 Iterated map 19—20 22 157—163 444—445 Jacobian matrix 97—100 131 Julia set 65 353 Jump time 384—386 KAM (Kolmogorov — Arnold — Moser) surface and tori 291—292 294 306—307 440—441 500—501 KAM (Kolmogorov — Arnold — Moser) theorem 290—291 Kaplan — Yorke conjecture 382 Karhunen — Loeve decomposition 479 Kneading sequence 174 Koch curve 346—347 Koch snowflake 347 Kolmogorov — Sinai (K-S) entropy 335—341 404 419 Kolmogorov — Sinai (K-S) entropy, relationship to Lyapunov exponents 339 419 Lagrangian viewpoint 278 453 Landau scenario 215 Laplace 38—39 Laplacian 452—455 Laser dynamics 598—604 Legendre transformation 397 410 Lifted state space 109 Limit cycle 87 96 102—106 Limit cycle, attracting (stable) 103 111 132—133 Limit cycle, repelling (unstable) 103 111 132—133 Limit cycle, stability 102—106 132 Linear systems 4—6 Liouville theorem 277—279 Local bifurcations 121—122 Logistic differential equation 86 Logistic map 19—26 47—57 192 Logistic map, bifurcation diagram 25 Logistic map, chaotic bands 24—25 Logistic map, invariant measure 332 Logistic map, Misiurewicz point 181 Lorenz model 27—37 145 547—558 Lorenz, E. 27 38
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