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Cambridge University Press 978-0-521-83912-9 - Chaotic Dynamics: An Introduction Based on Classical Mechanics Tamas Tel and Marton Gruiz Index More information

Index

accumulation point, 150, 221 fractal, 9, 12 acoustics, 301 basin boundary metamorphosis, 222 advection, 305 basin of attraction, 9, 11, 71, 78, 107, 121, 141, 144, 146, chaotic, 17, 306 197, 221, 225, 296, 308 advection map, 307, 309 Bernoulli map, 150 angle of deflection, 265 bifurcation, 73 aperiodic motion, 21 crisis, 217 area contraction rate, 105 distorted pitchfork, 75 Arnold, A., 23 pitchfork, 73, 185 Arnold diffusion, 319 bifurcation diagram, 74, 152, 185, 187, 218, 220 asteroid, 283, 284, 286 bifurcation point, 73 atmosphere, 18, 293, 316 billiard, 260 attractor, 51 concave, 260 chaotic, 8 open, 265, 301 limit cycle, 78, 323 Sinai, 260 node, 64, 86, 104 stadium, 260 point, 61 biochemical reactions, 302 simple, 7, 9, 71, 78, 297 Birkhoff, G., 23 spiral, 62, 69, 77, 83, 86, 98, 104 bistability, 67 torus, 319 bistable system, 67, 70, 297 attractor enlargement, 219 bouncing autonomous map, 94 between discs, 15, 270 autonomous system, 80, 299 on a double slope, 14, 242, 318 average Lyapunov exponent, 156, 166, 170, 173, 177, on a vibrating plate, 174, 318 205, 206, 214, 277, 285, 338 boundary crisis, 219 Brownian motion, 21, 168, 240 baker attractor, 121 buckling rod, 292 baker map, 113, 200, 213, 233 butterfly effect, 159 area preserving, 230 asymmetric, 129 Cantor, G., 47 open, 193 Cantor cloud, 32, 196 open area preserving, 273 asymmetric, 36, 46, 199, 200 open asymmetric, 199 Cantor filament, 35, 121, 160, 197 open symmetric, 193 asymmetric, 45 baker saddle, 200, 204 fat, 39 asymmetric, 199 Cantor set, 32, 41, 196, 221 ball in a vessel, 250 asymmetric, 34 basic branch, 122, 134, 137, 142, 198 fat, 39 basin boundary, 9, 71, 79, 88, 140, 144, 147, 171, two-scale, 34 214, 225, 307 cat map, 232

387

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-83912-9 - Chaotic Dynamics: An Introduction Based on Classical Mechanics Tamas Tel and Marton Gruiz Index More information

388 Index

catastrophe, 75 crisis bifurcation, 218 centrifugal force, 248, 280 cycle point, 127, 154 chaos, 3, 58, 254 conservative, 22, 46, 227, 259, 264 determinism, 168 definition of, 27 deterministic chaos, 168 deterministic, 168 deterministic system, 168, 240 dissipative, 22, 46, 113, 171, 191 devil’s staircase, 221 macroscopic, 258 differential equation, 106 microscopic, 256, 258 diffusion, 225, 239, 315 molecular, 319 chaotic, 239 permanent, 22, 113, 191, 227 diffusion coefficient, 240 transient, 9, 22, 191, 210, 264 dimensionless equation, 326 chaotic advection, 17, 18, 306, 318 dimensionless form, 325 environmental significance of, 315 direct product, 35, 39, 46 chaotic attractor, 8, 22, 45, 121, 139, 171, 178, 180, 214, 221, discrete dynamics, 94 259, 302 dissipation, extremely strong, 149 of permanent winter, 297, 299 dissipative system, 7, 19, 22, 46, 56, 61, 82, 194 chaotic band, 12, 13, 15, 22, 46, 234, 239, 241, 258, 260, 278 driven harmonic oscillator, 95, 177, 251 chaotic dynamics, 22 driven non-linear oscillator, 178, 328 chaotic lifetime, 192, 202 driven oscillator, 4 chaotic motion, 3 driven pendulum, 7, 8, 9, 10, 179, 180 chaotic saddle, 10, 22, 46, 191, 196, 198, 201, 209, 210, 212, driven system, 90 214, 220, 264, 271, 272, 277, 286, 308, 316 driving, 7, 95, 299 chaotic scattering, 16, 22, 46, 264, 277, 301, 307, 316 driving force, 4 chaotic set, 22, 46, 221, 303, 331, 332, 338 droplet dynamics, 305, 311, 315 chaotic transient, 11, 22, 191, 202, 221 dye, 304, 316 chemical reaction, 276, 302, 303 dynamical entropies, 319 in flow, 316 dynamical instability, 156 Christmas-tree ornaments, 16 dynamical similarity, 326 climate variability, 293, 297 closed flow, 309 Earth, 3, 279, 286 coiled structures magnetic field of, 301 serpentine, 293 earthquake, 301 telephone cord, 293 economy, 302 concert hall acoustics, 301 eigenvalue, 85, 97, 103, 134 conservative system, 12, 15, 19, 22, 46, 83, 227 eigenvector, 87, 97, 104, 134 autonomous, 242 electronic circuit, 301, 303 continued fraction, 253 elliptic island, 234, 239, 258 continuity equation, 183 elliptic point, 60, 86, 104, 234, 237, 257 continuous dynamics, 94 energy, 294 control, 302 ensemble forecast, 300 OGY algorithm, 303 ensemble of particles, 81 controlling chaos, 225, 303 epidemics, 320 Coriolis force, 248, 280 equilibrium state, 52 coverage of objects, 28 stable, 59 cream in coffee, 17 unstable, 52 crisis, 214, 216, 217, 218, 222 ergodic system, 261 boundary, 219, 222 ergodicity, 261 internal, 219, 222 error, 172 other types of, 219 escape from potential well, 226

Index 389

escape rate, 192, 196, 202, 207, 210 Hénon-type map, 148 escape regions, 273 heteroclinc point, 259 escape route, 273 high degree of freedom system, 261, 263, 319 Euler method, 329 higher-dimensional phase space, 319 experiment, observation of chaos in, 318 higher-order cycle, 92, 107, 125, 144, 154, 197, exponential decay, 192 198, 241 homoclinic point, 128, 172, 176, 198, 206, 241, fat fractal, 38, 39, 46, 154, 258, 278 258, 259 fat fractal exponent, 38, 154, 259 homogenously chaotic system, 260 Feigenbaum, M., 23 Hopf, E., 23, 233 Feigenbaum constants, 151 horseshoe, 222 fixed point, 52, 59, 65, 115 horseshoe map, 213 elliptic, 60 humming-top, 287, 290 hyperbolic, 53 hydrodynamics, 187, 304 marginal, 86, 104 hyperbolic cycle, 126 node, 64 hyperbolic point, 53, 57, 69, 83, 86, 88, 99, 104, 115, 134, spiral, 62 185, 200, 237, 257, 258 flour beetle, 302 Hyperion, irregular rotation of, 285 flow, 80, 82, 94 fluid, 304 image point, 93 three-dimensional, 92, 109, 175 impact parameter, 265 two-dimensional, 80, 85 information content, 43 food chain, 302 information dimension, 42, 44, 46, 170, 173, 177, 204, 207, forecast, 300 208, 277, 341 fractal, 6, 16, 17, 23, 260, 316 partial, 165, 205 fat, 38 instability, 22, 51, 58, 67 multi-scale, 33 instability exponent, 57, 86 one-scale, 32 integrable system, 254, 258 thin, 38 intermittency, 318 fractal basin boundary, 9, 12, 22, 46, 221, 222, internal crisis, 219 297, 302 internet and chaos, 302 fractal dimension, 29, 46, 122, 160, 173, 224, 341 invariant curve, 99, 107, 128, 251, 254, 258 partial, 37, 196, 197, 203 invariant set, 171 fractal distribution, 40, 43, 46, 164, 204, 214 invariant surface, 250 fractal escape boundaries, 273 inverse map, 102, 107, 108, 116, 131 fractal structure, 21, 173 invertibility, 84, 103 friction, 56 invertible dynamics, 84 frictional system, 22 invertible map, 98, 106 frictionless system, 12, 14, 22 irrational torus, 253 Frobenius–Perron equation, 163 irregularity, 116, 173, 260 irreversibility, 82, 262, 263 golden mean, 253 island elliptic, 234, 258 harmonic oscillation regular, 46, 258, 277, 310 damped, 62 iteration, 93 overdamped, 63 heart, 302, 303, 320 Jacobian, 105, 106, 116, 134, 157, 168, 177, 193, 230 Helium atom, 277 Jupiter, see under planets Hénon attractor, 148 Hénon map, 147 KAM theorem, 23, 253, 255, 258, 260, 286

390 Index

KAM torus, 255, 258, 277, 310 map, 20, 93, 94, 106, 322 Kantz–Grassberger relation, 207 advection, 307, 309 Kaplan–Yorke relation, 169, 177, 209 area preserving, 230 kicked free motion, 275, 325 baker, 129 kicked oscillator, 100, 131, 199, 221, 222, 324 Hénon, 148 kicked oscillator map, 102 Hénon-type, 148 kicked rotator, 234, 238, 240 invertible, 98, 109 kicked system, 95 kicked free motion, 325 Kirkwood gap, 285 kicked oscillator, 102, 131, 324 kneading dough, 233 kicked rotator, 235 Koch, H. von, 47 Lozi, 148 Koch curve, 26, 30, 46 non-autonomous, 303 Kolmogorov, A., 23, 47 non-invertible, 109 Kovalevskaia, S., 23, 290 one-dimensional, 109, 164 Kovalevskaia constant, 290 parabola, 145 Poincaré,21, 22, 189, 229 Lagrange point, 283, 286 roof, 140 laser, 275, 301, 303 sawtooth, 135 lifetime scattering, 269 of chaos, 192, 301 standard, 235 of transition complex, 276 stroboscopic, 21, 22, 93, 324 lifetime distribution, 209, 268, 272 two-dimensional, 103, 104, 105, 109, 169, limit cycle, 88, 92, 100, 175, 295, 303, 323 171, 175 attractor, 78, 96 marginal fixed point, 86, 104 hyperbolic, 99 Mars, see under, planets periodic, 176 Menger sponge, 33 unstable, 99, 218, 324 mirroring balls, 16 line segment, 155, 202, 337 mixing, 233, 261, 305, 316, 318 linear approximation, 51 mixing system, 261 Liouville’s theorem, 82, 83 molecular chaos, 319 local Lyapunov exponent, 156, 167, 203, 301, 338 Moon, 26, 280, 283, 286 logistic map, 108, 150 surface of, 26, 30 complex, 108 Moser, J., 23 Lorentz gas, 260 motion on a relief, 58, 66 Lorenz E., 23, 47, 159, 187, 293 multi-fratal, 43 Lorenz model, 187, 301 multi-scale fractal, 33 Lorenz’s model of global circulation, 293 musical instrument, 301 Lozi attractor, 148 Lozi map, 148 nanotechnology, 301 Lyapunov exponent, 118, 196, 231, 316, 318 natural distribution, 46, 161, 164, 167, average, 156, 168, 170, 205, 338 172, 214 local, 156, 203, 338 of the baker attractor, 162, 166 of chaotic attractor, 161 magnetic pendulum, 8, 224 of kicked oscillators, 165 Mandelbrot, B., 47 of permanent winter, 298 Mandelbrot set, 108 of saddle, 203 manifold of the water-wheel attractor, 190 numerical determination of, 73 of the water-wheel saddle, 219 stable, 70, 107 natural frequency, 60, 86 unstable, 71, 107 natural measure, 161

Index 391

noble numbers, 253 perturbation, 255, 257 node attractor, 64, 86, 104 phase portrait, 71, 78, 79, 87, 141, 145, 147, 171, 216, 229, node repellor, 86, 104 234, 238, 246, 269, 310 noise, 21, 119, 240, 262, 319 phase space, 19, 45, 53, 55, 60, 66, 69, 79, 80, 91, noise-induced chaos, 225 184, 262, 305 non-attracting set, 191 at least three-dimensional, 109 non-autonomous system, 91 phase space contraction rate, 81, 86, 177, 228 non-hyperbolic effects, 318 phase space volume, 81, 83 non-invertible map, 108 pinball machine, 15 non-linear behaviour, 51 planets, 228, 280 non-linear force law, 5, 66 Jupiter, 280, 283, 285 non-linear oscillator Mars, 285 driven, 5 Pluto, 285 non-linear system, 109 Saturn, 284 non-linearity parameter, 132, 150 plasma, 301 numerical error, 329 Pluto, see under, planets numerical simulation, 4, 172, 236, 332 Poincaré,H., 23, 280 numerical solution, 325, 329 Poincarémap, 21, 219, 229, 268 Poincaréportrait, 229, 248, 249, 283, 289 observed perimeter, 25, 29 Poincarésection, 229 observed surface, 25, 29, 37 Poincaré–Bendixsontheorem, 88 observed volume, 37 Poincaré–Birkhoff theorem, 257 ocean, 316 point attractor, 61 one-dimensional map, 149, 164, 201, 217 pollution, 16, 304, 316 one-disc problem, 266 potential, 58, 65, 76, 281 one-scale fractal, 32 double-welled, 68 open baker map, 193 pre-image, 102 open flow, 307, 316 prediction time, 158, 160, 205, 262 oscillation principal moment of inertia, 286, 293 driven, 4 probabilistic concepts, 21, 23 harmonic, 60 probability distribution, 40, 160, 203 program, 332 parabola amplitude, 101, 132 diffeq, 334 parabola attractor, 146, 214, 220, 223 map, 332 parabola map, 145, 152, 220 parabola saddle, 204 quasi-periodic motion, 237, 252, 254, 258, 284, 313 parameter dependence, 150, 214 partial dimension, 37 rain, 181 partial fractal dimension, 37, 196, 197 random motion, 58 partial information dimension, 165, 204 random walk, 168, 239, 331 pendulum, 327 rational torus, 253 clock, 3 reaction in flow, 316 driven, 7, 9, 179, 224, 318 reduced mass, 281 magnetic, 8, 224, 318 regular island, 13 spring, 246, 318, 327 regular motion, 3, 45, 51, 313 period-doubling cascade, 150, 151, 218 repellor, 108 periodic orbit, 92, 125, 129, 154, 172, 210, 229, 259, 268, node, 86, 104 271, 274, 276, 303, 318 spiral, 86, 104 periodic window, 153, 219 resolution, 25, 29 permanent chaos, 22, 46, 113, 191 resonance, 5, 286, 322

392 Index

resonant torus, 254, 256, 258 stability analysis, 65 Rössler model, 190 stability matrix, 85, 103, 116, 133, 144, 145, 236 roof amplitude, 132 stable curve, 55 roof attractor, 141, 337 stable equilibrium point, 59 roof map, 140, 215, 216 stable manifold, 70, 79, 88, 99, 100, 105, 107, 128, 140, 144, roof saddle, 204, 337 146, 175, 198, 209, 212, 221, 222, 242, 259, 271, 273, round-off error, 262, 331 277, 303, 308 Runge-Kutta method, 330, 334 stable state, 59, 66, 95 standard map, 234, 236 saddle state of a system, 19 chaotic, 10, 191 strange attractor, 8 parabola, 200 stroboscopic map, 21, 93, 315 roof, 200, 337 structural stability, 57 sawtooth, 200 Sun, 3, 280, 283, 286, 294 water-wheel, 219 swinging on a pulley, 12, 248 saddle point, 53 Saturn, see under planets tank with two outlets, 17, 306 sawtooth amplitude, 132 tent map, 150 sawtooth attractor, 136 thermal convection, 187 sawtooth map, 135, 215 thermal equilibrium, 302 sawtooth saddle, 204 thin fractal, 38 scattering, 16, 264 three-body problem, 279, 283 scattering function, 265, 268, 272 three-cycle, 125, 270 scattering map, 269, 275 three-disc problem, 15, 270, 271, 272 scattering region, 265, 273 time reversal invariance, 84, 259, 277 self-similarity, 26, 27 time step, 329 sensitivity to initial conditions, 4, 6, 14, 56, time-reversed dynamics, 71, 84, 102 156, 262 top separatrix, 71, 88, 254 asymmetric, 287 set of measure zero, 38, 121, 196 Kovalevskaia, 291 shear flow model, 309 spinning, 286 shimmying wheel, 292 symmetric, 287, 290 ship capsizing, 72, 226 topological entropy, 127, 155, 167, 173, 177, 197, 202, 205, Sierpinski gasket, 32 214, 221, 231, 337 Sierpinski tower, 32 torus, 250, 251 simple attractor, 7, 9, 11 irrational, 253 simple system, 80 rational, 253 skiing on a slope, 76 resonant, 254, 256, 257 Solar System, 23, 228, 280, 285 torus attractor, 319 see also under planets trailer, lateral motion in, 292 spacecraft, 228, 279 trajectory, 19, 53, 60, 62, 69, 80, 92, 94 spatial patterns, 320 transient chaos, 9, 11, 16, 22, 46, 191, 210, 264, 277, 297, spatio-temporal chaos, 320 301, 302, 307 spinning-top, 23, 286, 289, 293 transition complex, 276 spiral attractor, 62, 86, 98, 104 transport barrier, 310 spiral repellor, 86, 104 transport processes, 225 spreading of pollutants, 16, 17, 304 turbulence, 320 spring pendulum, 246 two-cycle, 124, 136, 137, 197, 238, 269 stability, 66 two-disc problem, 266 of fixed points, 85, 103, 133, 134, 184, 236 two-scale Cantor set, 34

Index 393

uncertainty exponent, 223 blinking, 18 universality, 151 street, 316 unpredictability, 4, 6, 12, 21, 117, 158, 173, 260 vortex dynamics, 312 unstable curve, 55 chaotic, 313 unstable manifold, 71, 77, 88, 99, 100, 105, 107, 122, 123, 139, 142, 146, 172, 176, 198, 211, 212, 231, 242, 273, water-wheel, 181, 192, 217, 225 277, 305, 307, 316 water-wheel attractor, 188, 189 unstable state, 7, 65, 98, 323 water-wheel saddle, 219 west wind, 294 van der Pol oscillator, 83, 89 winding number, 250, 255 von K´arm´anvortex street, 316 vortex Yorke, J., 23