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Lorenz E.N. — Essence of Chaos
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Íàçâàíèå: Essence of Chaos
Àâòîð: Lorenz E.N.
Àííîòàöèÿ: Chaos Surrounds us. Seemingly random events - the flapping of a flag, a storm-driven wave striking the shore, a pinball's path - often appear to have no order, no rational pattern. Explicating the theory of chaos and the consequences of its principal findings - that actual, precise rules may govern such apparently random behavior - has been a major part of the work of Edward N. Lorenz. In The Essence of Chaos, Lorenz presents to the general reader the features of this "new science," with its far-reaching implications for much of modern life, from weather prediction to philosophy, and he describes its considerable impact on emerging scientific fields. Unlike the phenomena dealt with in relativity theory and quantum mechanics, systems that are now described as "chaotic" can be observed without telescopes or microscopes. They range from the simplest happenings, such as the falling of a leaf, to the most complex processes, like the fluctuations of climate. Each process that qualifies, however, has certain quantifiable characteristics: how it unfolds depends very sensitively upon its present state, so that, even though it is not random, it seems to be. Lorenz uses examples from everyday life, and simple calculations, to show how the essential nature of chaotic systems can be understood. In order to expedite this task, he has constructed a mathematical model of a board sliding down a ski slope as his primary illustrative example. With this model as his base, he explains various chaotic phenomena, including some associated concepts such as strange attractors and bifurcations. As a meteorologist, Lorenz initially became interested in the field of chaos because of its implications for weather forecasting. In a chapter ranging through the history of weather prediction and meteorology to a brief picture of our current understanding of climate, he introduces many of the researchers who conceived the experiments and theories, and he describes his own initial encounter with
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Ãîä èçäàíèÿ: 1996
Êîëè÷åñòâî ñòðàíèö: 240
Äîáàâëåíà â êàòàëîã: 14.05.2008
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Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
Adams, John Couch (1819-1892) 113
Almost-periodicity: absence of, and chaos 19—20 142
Almost-periodicity: in Hamiltonian systems 65 67 67
Almost-periodicity: meaning of 18—19
Analogues 85—86
Arrhythmias 15—16 127 149
Asymptotic solutions 118 199
Atmosphere as dynamical system 78 79.
Attracting sets: determination of points on 59—60 60.
Attracting sets: meaning of 59
Attractors: absence of, in Hamiltonian systems 61—62
Attractors: determination of points on 43—45 44 53
Attractors: graphical representation of 41—43
Attractors: in dishpan experiments 89—91
Attractors: meaning of 39—41
Attractors: multiple 55
Attractors: of board model 53—55 54 197—198 198
Attractors: of global weather system 50 97 98 99—100 130
Attractors: of sled model 45—48 46 47 49 197 198
Attractors: of specially designed systems 153 155 155—156 156 157 186 187
Bamsley, Michael Fielding 173
Baseballs 111—112
Basin boundaries: in board model 55—59 56
Basin boundaries: meaning of 55
Basins of attraction: in board model 55—59 74
Basins of attraction: meaning of 55
Bernoulli shifts 204
Bifurcations: in board model 70—76 72 73 75 76 147
Bifurcations: in dishpan experiments 89 147
Bifurcations: in logistic equation 194
Bifurcations: meaning of 69
Bifurcations: period-doubling 71 72 73 74—76 76
Bifurcations: resulting from stability changes 69—70
Bifurcations: saddlenode 70
Birkhoff's "bad" curves 121—124 122 123
Birkhoff, George David (1884-1944) 121 122 123 124 125 127 128 142
Bjerknes, Vilhelm Friman Koren (1862-1950) 95
Board model: attractor of 54 177 197—198 198
Board model: basin boundaries in 55—59 56
Board model: bifurcations in 70—76 72 73 75 76
Board model: detection of chaos in 32 33 34 35—37 36
Board model: determination of attractor of 53—54
Board model: formulation of 26—32
Bradbury, Ray 15
Brahms, Johannes (1833-1897) 150
Business cycles 149 151
Butterfly attractor 14 15 139—141 140 141 145 146 150 177 188
Butterfly effect 14--15 181—184
Butterfues: as symbols of chaos 13—15
Butterfues: as weather modifiers 85 153 181—183
Cantor sets 50 53 176—177 178 194 204
Cantor, Georg (1845-1918) 50
Capacity 168 196
Card shuffling 7 9
Cartwnght — Littlewood attractor 124 124—125 167
Cartwnght, Dame Mary Lucy 123
Cellular automata 174—175
Chaos: "make your own" 151—157 152 153 154 155 See
Chaos: acquisition of technical meaning 20—21 120
Chaos: and atmospheric predictability 102 103—106 142—143
Chaos: awareness of 113—114 120—121 125 128—129 146—147
Chaos: detection of 16 20 25—26
Chaos: in dishpan experiments 89 92
Chaos: in global circulation models 95 102
Chaos: in the atmosphere 79 86 94 108—109
Chaos: meanings of 3—4
Chaos: perceived as randomness 118—119 157—159
Chaos: recognition of, by Poincare 118—120
Chaos: refinements of technical meanings 5—6 7 8 23—24
Chaos: routes to and from 69—70
Chaos: symbols of 13—15
Chaos: technical meaning of, in this volume 4
Chaotic seas 64 66—68 68 147
Charney, Jule Gregory (1917-1981) 98 100 103 104 142
Chirikov, Bons 48 53 191
Climate, S. 50 79
Cloud physics 84. See also Clouds; Rain; Snow
Clouds 81—89 passim
Coefficient of function 28 39 189—190
coin tossing 6—7 9 12 14
Compactness: meaning of 16—17
Compactness: of board model 35 37
Compactness: of global weather system 82 85—86
Compactness: significance of 18—20
Complete randomness 7—8
complexity 4 163 164 165 166 166 167 170
Computation, manual 114 117 128
Computer graphics 43 185
Computers: and awareness of chaos 128—129.See also LGP-3O computer
Computers: and weather forecasting 82 95 98—106
Computers: as equation solvers 26 28 32 114 131 156 157 171
Conservative systems 61. See also Hamiltonian systems
Convection 137 143 145 148
Cross sections 43—56 passim. See also Double cross sections; Poincare sections
Cusps 56 57—58 59
Damping time 28 30
Day, length of 78 94
Determinism 7 8 9 157 159
Devaney, Robert 194
Dewey, Thomas Edmund (1902-1971) 77
Difference equations: as basis for dynamical systems 127. See also Logistic equation; Mappings; Population dynamics
Difference equations: derived from differential equations 13 47—48 121
Difference equations: meaning of 12
Differential equations: as basis for dynamical systems 121 142.
Differential equations: general solutions 111—112 117 139
Differential equations: in sled model 47
Differential equations: meaning of 13
Differential equations: solution curves 117—118
Dimensionality 168—177 passim 178 196—198
Discontinuous mappings 151—152
Dishpan experiments: apparatus for 87—89 88 90 92
Dishpan experiments: flow patterns in 89—92 90 91 93 94 95
Dishpan experiments: implications of 92—94
Disk dynamos 148
Dissipative systems: atmosphere and ocean as 78
Dissipative systems: meaning of 51
Dissipative systems: versus Hamiltonian systems 65 67
Double cross sections 56—57 58—59
Doubling times 104—106 182—183
Doubly asymptotic solutions 118 199
Duffing oscillator 167
Dynamic meteorology 82—87 passim 95 130
Dynamical systems: families of 28 See
Dynamical systems: meaning of 8
Economy 149
Ellipses: as attractors 42 61
Ellipses: as images of circles 50—53 52
Ellipses: as planetary orbits 112 114
Ellipsoids, as images of spheres 55 195—196
Energy 57 62—64 68 99
Equations: of dynamical systems 12—13
Equations: of global weather system 95—96 98.
Equilibrium 22—23. See also Stable equilibrium; Unstable equilibrium
Errors: in weather forecasting 104 105 106 182—184
Errors: in weather observations 85 96—104 136 182—183
European Centre for Medium Range Weather Forecasts 101—102
European Centre model 101—102 104—105
Faller, Alan Judson 92 145
Families of systems 28 62 69
Feigenbaum, Mitchell Jay 194
Filtered equations 99 100 104 132
Flags, flapping, as chaotic systems 5 16 17 20 24
Flows: mappings derived from 13 47—48 121.
Flows: meaning of 13
Fractal leaves 173 176 177
Fractal trees 171—173 172 174 175
Fractal triangles 169 170 170—171 173—174
Fractality 4 169—178 170 171 178. Fractal
Franklin, Philip (1898-1965) 142
Free will 158 159—160
Friction: in board model 27 28 57 74
Friction: in dissipative systems 51
Friction: in pendulums 5 61
Friction: in pinball machine 10 17
Friction: in real sliding systems 152. See also Coefficient of friction; Damping time
Fronts 92
Full chaos: in Birkhoff and Cartwright — Littlewood systems 122—125
Full chaos: meaning of 21—22
Full chaos: recognition of by Poincare 118—119
Full chaos: versus limited chaos 126—127
Fultz, Dave 87 89 92
Gleick, James 4 14 15 21 179
Global Atmospheric Research Program 80 103 110 142—143 181 184
Global circulation models 99—105 passim 109—110 142—143
Global weather system 80—82 85
Golfballs 16—17 112
Graphics 43 185
graphs 18 37—38 41
Gravity, force of: in atmosphere 86 93
Gravity, force of: in dishpan experiments 86 87
Gravity, force of: in ski-slope model 28
Gravity, force of: in solar system 111 113
Guckenheimer, John 49
Haken, Hermann 148
Hamilton, Sir William Rowan (1805-1863) 61
Hamiltonian board model: chaotic seas and islands in 64 65 65—68 67 68
Hamiltonian board model: detection of chaos in 63 64—65
Hamiltonian board model: formulation of 62—64
Hamiltonian systems: as atmospheric models 99
Hamiltonian systems: chaos and periodicity in 64—68 116
Hamiltonian systems: Lyapunov exponents of 195. See also Hamiltonian board model; Standard map; Volume-preserving systems
Hamiltonian systems: meaning of 61—62
Hausdorff dimension 168
Hausdorff, Felix (1868-1942) 168
Heartbeat 15—16 24 127 149
Hide, Raymond 89 90 91 92
High pressure systems 82 83 96—97
Hill's reduced problem 114—117 115 116 133 192—193
Hill, George William (1838-1914) 114 116
Homoclinicity 118 199—204
Horseshoe mappings 125—126 126 127 156 202—204
Howard, Louis Norberg 143
Humidity 80—81 101 105
Initialization 100 102
Instrumentation 84
Invariant sets 59 123—124
Invertible mappings 153
Izrailev, Felix 48 53
Jacobi integral 193
Japanese attractor 165 167
Jet streams 82 89 92—93 100 105
Jupiter 112
Kaplan — Yorke dimension 196
Kaplan, James 196
Keefe, Douglas 150
Knot theory 38 146 147
Kolmogorov, Andrei Nikolaevich (1903-1987) 168
Krakatau 106—107
Krishnamurti, Ruby 143
Kutta, Wilhelm (1867-1944) 185
Laboratory models 26 86—87 See
Laplace, Pierre Simon de (1749-1827) 159
Lasers 148
Leaves, falling, as chaotic systems 5 158
Length of day 78 94
LeVerrier, Urbain Jean Joseph (1811-1877) 113
LGP-3O computer 132—136 137
Li, lien Yien 20—21 120 145
Limited chaos: in Birkhoff and Cartwright-Littlewood systems 122—125
Limited chaos: in horseshoe mapping 125—126 204
Limited chaos: meaning of 21—22
Limited chaos: recognition of, by Poincare 118
Limited chaos: versus full chaos 126—127
Linear predichon 130—134
Linearity 161—162
Littlewood, John Edensor (1885-1977) 123
Lockett, Carolyn 150
Logistic equation 80 147—148 194—195
Lorenz attractor See Butterfly attractor
Lorenz equahons 188 See
Lorenz, Edward Norton 14—15 69 128 130 136 150—151.
Low pressure systems 82 83 92 96—97 98
Lyapunov exponents 177 195—197
Lyapunov numbers 195 197
Lyapunov stability 142
Lyapunov, Aleksandr Mikhailovich (1857-1918) 142
Malkus, Willem van Rensselaer 143
Malone, Thomas Francis 130 137
Mandelbrot set 163—165 164 167 195
Mandelbrot, Benoit 169 170 173 177
Manifolds: as components of attractors 48 196
Manifolds: stable and unstable 99—202 200 201 203 203—204
Many-body problem 111
Mappings: as generators of chaos 151 153—157
Mappings: derived from flows 13 47—48 121
Mappings: discontinuous 151—153
Mappings: meaning of 12
Mappings: noninvertible 151 156 157 186 See
Markov, Andrei Andreevich (1856-1922) 142
Markov, Andrei Andreevich (son of A.A.M.) 142
Massachusetts Institute of Technology 130 137 142 143
Mathematical models, as approximations 5—6 10 17—18 27—28 35 45 87 95 112.
May, Robert McCredie 147 148
Menlees, Philip 15
Models See Laboratory models; Mathematical models
Molecules 120
Monte Carlo forecasting 102—103 106
Moon 79
MUSIC 149—150
Nahonal Meteorological Center 102
Nemytskii, Viktor Vladimirovich 142
Neptune 111 113 114
Newton's Laws 26—27 82 111 148 149 189
Newton's method 187—188
Newton, Sir Isaac 111
Noninvertible flows 152—153
Noninvertible mappings 151 157 156 186
Nonlinearity 4 161—163
Numerical integration 96 112 185—188
Numerical weather prediction 95—106 passim 131
Orbits, in phase space 42 117
Orbits, planetary 111—116 passim 128
Oscillations, in atmosphere 96—102 passim. See also Quasi-biennial oscillation
Oscillators: Duffing 167
Oscillators: van der Pol 123
Parabolas 111—112
Parameterizahon 10
Pencils, standing, as unstable systems 22 119
Pendulums 4 5 39 42—43 51 61 162
Period doubling: in board model 71 72 73 74—76 76
Period doubling: in logistic equation 194
Period doubling: meaning of 71
Period doubling: prevalence of 71
Periodicity: absence of, and chaos 18—20 26 35—37 133 136 142
Periodicity: bifurcation to and from 71—74
Periodicity: hypothetical, in atmosphere 108—110 See
Periodicity: in atmosphere 86
Periodicity: in dishpan experiments 90—91
Periodicity: in Hamiltonian systems 67—68 115—116
Periodicity: introduced by Poincare 117
Periodicity: meaning of 18—19
Periodicity: of pendulum 42—43
Periodicity: of ski-slope model 43—44 46—47
Perturbation method 112—113
Phase space, meaning of 41—42
Phillips, Norman Alton 100 139
Pinball machines: as dynamical systems 9—10 13
Pinball machines: chaotic behavior in 10 11 12 18 19 20 62
Pinball machines: elongated 17—18 19 20
Platzman, George William 99
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