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Smith L.A. — Chaos: A Very Short Introduction

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Название: Chaos: A Very Short Introduction

Автор: Smith L.A.

Аннотация:

Chaos exists in systems all around us. Even the simplest system can be subject to chaos, denying us accurate predictions of its behavior, and sometimes giving rise to astonishing structures of large-scale order. Here, Leonard Smith shows that we all have an intuitive understanding of chaotic systems. He uses accessible math and physics to explain Chaos Theory, and points to numerous examples in philosophy and literature that illuminate the problems. This book provides a complete understanding of chaotic dynamics, using examples from mathematics, physics, philosophy, and the real world, with an explanation of why chaos is important and how it differs from the idea of randomness. The author's real life applications include the weather forecast, a pendulum, a coin toss, mass transit, politics, and the role of chaos in gambling and the stock market. Chaos represents a prime opportunity for mathematical lay people to finally get a clear understanding of this fascinating concept.

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Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 180

Добавлена в каталог: 22.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

AC Map      42—43
Accountability      125—126
Accuracy      125—126
Advection models      132—133
Almost every      34 163
Analogue models      133 134
Area-perimeter method      81
Attractors      35—39 45—47 163
Attractors, chaotic      69—71 164
Attractors, Henon      69—71
Attractors, Henon — Heiles      72
Attractors, Lorenz      66 67
Attractors, Moore — Spiegel      67 69—71
Attractors, strange      84—85 168
Auto-Correlation Function (ACF)      88
Babbage, Charles      123
Baker's Apprentice Maps      98 99—101
Baker's Map      98—101
Barometer      6
base two      88—89
Basin of attraction      163
Bayesians      126
Bell-shaped curve      8 9 52 114—115 155
Ben Dahir, Sissa      22—23
Bifurcation diagram      60—62
Binary notation      88—89
Blood cells      71
Borders/boundaries      80
Bradbury, Ray      1 5—6
Brillouin, Leon      104
Burns effect      15 142 146 163
Burns' Day storm      10—16 139—142
Butterfly effect      1 5—6 15 164
Cantor, Georg      77
card tricks      108 108—110
Causation      88
Chaotic attractors      69—71 164
Cheat with the Ace of Diamonds (de la Tour)      19—21
Chess      22—23
Climate modelling      143—146 159—160
Clouds      81
Coastlines      80
Commerce      146—147
computer simulations      34 43—44 90—91 107—110 117—120 135—136
Computers      88—91 107—110
Conservative dynamical systems      164
Correlation      88
Darwin, Charles      4—5
Data assimilation      122
Data-based models      117—120 132—134
De Morgan, A.      76 80
Delay equations      71
Delay reconstruction models      117—120 133 164
Delay-embedding state space      116—120
Determinism      1 3 42 88 90 107 158 164
Digitally periodic loops      107—110
dimensions      65
Dimensions, dimension estimates      115—116
Dissipative chaos      66—71 83
Dissipative dynamical systems      45 164
Doubling time      92—93 101 164
Duration of observations      120
Dynamic noise      55
Dynamical systems      33—35
Dynamical systems, computer simulations      34 43—44 90—91 107—110 117—120 135—136
Dynamical systems, creation of information      89—90
Dynamical systems, mathematical      33—52 89—90
Dynamical systems, physical      33—34 53—57 64 147—149 150—151 157—158
Earth's atmosphere/ocean system      148
Eddington, Arthur      17—18 138 159
Effectively exponential growth      93 164
Electronic circuits      148—149 150—151
Embedology      116—120
Energy sector      147 160
Ensemble forecasts      28—29 102 125—126 138—143 149 150—151 164—165
Ensemble weather prediction systems (EPS)      138—143
Epidemics      71
Errors      4
Errors, exponential growth of      27—28
Errors, forecast errors      29—31 106—107
Errors, observational uncertainty      4 160 166
Errors, representation error      106
European Centre for Mediumrange Weather Forecasts (ECMWF)      138 139—143 142—143
Evolution      5
Exponential growth      22—29 30 165
Exponential-on-average growth      93 94
Fair odds      152
Farmer, J.D.      116
Feigenbaum number      61—64
Fibonacci numbers      26—27
financial markets      146—147
Fitzroy, Robert      5 6 124 132—133
Fixed points      37 60—61 165
flows      65—66 165
Folding      29—32
Forecast errors      29—31 106—107
Forecasting      16 123—131 see
Forecasting weather      see Weather forecasting
Forecasting, accuracy and accountability      125—126
Forecasting, ensemble forecasts      28—29 102 125—126 138—143 149 150—151 164—165
Forecasting, model inadequacy      126—127
Forecasting, pandemonium      127—131
Fournier D'Albe, Edmund Edward      77—79
Fournier Universe      78—79 85
Fractal dimensions      80—81 85—86
Fractals      76—86 165
Fractals in physics      79—81
Fractals in state space      81—85
Fractals, solution to Olbers' paradox      77—79
Franklin, Benjamin      3—4
Full Logistic Map      37—39 39—42 88 96—97
Galton Boards      8 9 126—127
Galton, Francis      6—8
Geometric average      95—97 165
Grassberger, Peter      116
Great Storm of 1987      11
Growth, exponential      22—29 30 165
Growth, exponential-on-average      93 94
Growth, linear      23
Growth, stretching, folding and growth of uncertainty      29—32
Hamiltonian chaos      72
Henon attractor      69—71 86
Henon map      69—71 83—84
Henon — Heiles attractor      72
Hide, Raymond      148
Higher-dimensional systems      65—72 74
Higher-dimensional systems, Lyopunov exponents in      97—101
Hokusai, Katsushika      80
Implied probability      152
Indistinguishable state      165
Infinitesimal quantities      31 93—94 102—103 165
Information without correlation      88
Information, computers and      88—91
Information, content      91
Information, mutual      91—92
Integers      104—105
Iterated Function Systems (IFS)      43
Iteration      33 165
Judd, Kevin      115 152
Kepler, Johannes      77
La Tour, Georges De      19—21
Lacunae      85
Lagrangian chaos      65—66
Laplace's demon      3
Laplace, Pierre      3 4
Leading Lyapunov exponent      93
Least squares approach      114—115 155
Leonardo of Pisa      25
Leverrier, Urbain Jean Joseph      6 57 132—133
Limits      113

Linear dynamical systems      10 28 51 156 165
Linear growth      23
Logistic map      45 59—60 92 95 155
Logistic Map, attractors      46 48—49
Logistic Map, computer simulations      107—110
Logistic Map, universality      60—64
Long-range modelling      see models
Lorenz attractor      66 67
Lorenz system      66—69 101 157—158
Lorenz, Ed      6 66 74 133 145 148
Lothar/T1 storm of 1999      142—143
Low-dimensional systems      58—64 74
Lyapunov exponents      93—102 157 165—166
Lyapunov exponents in higher dimensions      97—101
Lyapunov exponents, positive exponents with shrinking uncertainties      101—102
Lyapunov time      166
Macbeth      124—125
Mach, E.      53
Machete's Moore — Spiegel circuit      117 118 149 150—151
Mandelbrot, Benoit      80 86
Maps      23 35—44 166 see
Markets      146—147
Mathematical dynamical systems      33—52 89—90
Mathematical dynamical systems, attractors      35—39 45—47
Mathematical dynamical systems, maps      35—44
Mathematical dynamical systems, parameters and model structure      44—45
Mathematical dynamical systems, statistical models of Sun spots      50—52
Mathematical dynamical systems, tuning model parameters and structural stability      47—50
Mathematical fractals      76 77
Mathematical models      15 58—75
Mathematical models, delay equations      71
Mathematical models, dissipative chaos      66—71
Mathematical models, exploiting insights of chaos      73—75
Mathematical models, Hamiltonian chaos      72
Mathematical models, higher-dimensional systems      65—72 74
Mathematical models, origin of mathematical term 'chaos'      65
Mathematical models, universality      60—64
Mathematics      20—21 34 55—56 159
Maximum likelihood      115
Maxwell, James Clerk      8—9
May, Lord      58—60
Medical research      71
Mercury      57
Meteorology      see Weather forecasting
Middle Thirds Cantor set      77 85—86
Middle Thirds IFS Map      43 81—83
Model inadequacy      57 111 126—127 130—131 152—153 160
Models      16 27 132—153 166
Models, climate      143—146 159—160
Models, data-based      117—120 132—134
Models, mathematical      see Mathematical models
Models, odds and probabilities      149—153
Models, parameters      see Parameters
Models, Phynance      146—147
Models, physical systems      147—149 150—151
Models, simulation      135—137
Models, weather forecasting      12—16 135—143
Moore — Spiegel attractor      67 69—71
Moore — Spiegel system      117 118 149 150—151
Moran — Ricker Map      41 59 60 64 95—96
Mutual information      91—92
NAG (Not A Galton) Board      127—131
Neptune      57
New Zealand      143
Newton's Laws      3
Newton, Isaac      73 79
Night sky, darkness of      77—79
Noise      53—54 166
Noise model      54 92 105 111 166
noise reduction      29
Noise, dynamic      55
Noise, exponential error growth      27—28
Noise, observational      4 55—57 92 106 111
Non-constructive proof      46 166
Nonlinearity      1 10 60 155 159—160 166
Nonlinearity, model parameter estimation      114—115
Numbers      104—105 155—156
Numerical weather prediction (NWP) models      135—137
Observational noise      4 55—57 92 106 111
Observational uncertainty      4 8 166
Observations      105—107
Observations and model states      154
Observations, duration of      120
Observations, operational weather forecasting      12—15
Odds      149—153
Olbers' paradox      77—79
Osceledec, V.      93
Packard, N.H.      116
Pandemonium      127—131 166
Parameters      24 166
Parameters and model structure      44—45
Parameters, best values for      113—115 154—155
Parameters, tuning      47—50
Perfect Model Scenario (PMS)      54 56 57 114 122 166
Period doubling      61—64
Periodic loops      43 50 61—64 65 86 167
Periodic loops, attractors      46 48—49
Periodic loops, digitally      107—110
Persistence models      132
Philosophy      20—21 35 53 57 154—161
Philosophy, burden of proof      157—158
Philosophy, complications      154—157
Philosophy, shadowing and the future      159—161
Phynance      146—147
Physical dynamical systems      33—34 53—57 64 157—158
Physical dynamical systems, models and      147—149 150—151
Physical dynamical systems, observations and noise      55—57
Physical fractals      76 77 79—81
Physics      20—21 34 56 159
Planets      57
Poe, Edgar Allen      4 8 77
Poincare section      71 167
Popper, Karl      125—126
Population dynamics      25—29 58—64 105—106
Predictability      16—18 51—52 123—131 167
Predictability, quantifying      91—97 101 see
Prediction Company (PredCo)      146—147
Probabilistic odds      152
probability      129—131 161
Probability, odds and      149—153
Procaccia, Itamar      116
proof      157—158
Proof, non-constructive      46 166
Quadrupling Map      36—37
Quantification      87—103
Quantification, computers and information      88—91
Quantification, dynamics of relevant uncertainties      102—103
Quantification, information without correlation      88
Quantification, Lyapunov exponents      93—102 157 165—166
Quantification, statistics for predicting predictability      91—97
quantum mechanics      54
Quartering Map      37 39 40 45 95
Quincunx      see Galton Boards
Rabbit Map      25—29
Random dynamical systems      42—44 54—55 89—90 167
Random number generators      44
Read, Peter      148
Real World      16—18
Real world, models and      147—149 150—151
Real world, science in      19—21
Recurrent trajectory      32 107 158 167
Representation error      106
Rice Map      22—24
Richardson, L.F.      77 79 80—81 135 137
Roulette      133—134 152
Ruelle, David      84
Sample-statistics      113 167
Self-similarity      76—77 78
Sensitive dependence      1—2 5—6 15 94 107 158 167
Shadowing      111 125 156 159—161 167

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