Chaos : an introduction to dynamical systems / Kathleen T. Alligood, Tim D. Sauer, James A. Yorke.
Series Textbooks in mathematical sciencesEditor: New York : Springer, c1997Descripción: xvii, 603 p. : ill. ; 24 cmTema(s): Differentiable dynamical systems | Chaotic behavior in systemsOtra clasificación: *CODIGO*Contents INTRODUCTION v I ONE-DIMENSIONAL MAPS I 1.1 One-Dimensional Maps [2] 1.2 Cobweb Plot: Graphical Representation of an Orbit [5] 1.3 Stability of Fixed Points [9] 1.4 Periodic Points [13] 1.5 The Family of Logistic Maps [17] 1.6 The Logistic Map G (x) = 4x( 1 — x) [22] 1.7 Sensitive Dependence on Initial Conditions [25] 1.8 Itineraries [27] Challenge 1: Period Three Implies Chaos [32] Exercises [35] Lab Visit 1: Boom, Bust, and Chaos in the Beetle Census [39] 2 TWO-DIMENSIONAL MAPS [43] 2.1 Mathematical Models [44] 2.2 Sinks, Sources, and Saddles [58] 2.3 Linear Maps [62] 2.4 Coordinate Changes [67] 2.5 Nonlinear Maps and the Jacobian Matrix [68] 2.6 Stable and Unstable Manifolds [78] 2.7 Matrix Times Circle Equals Ellipse [87] Challenge 2: Counting the Periodic Orbits of Linear Maps on a Torus [92] Exercises [98] Lab Visit 2: Is the Solar System Stable? [99] 3 CHAOS [105] 3.1 Lyapunov Exponents [106] 3.2 Chaotic Orbits [109] 3.3 Conjugacy and the Logistic Map [114] 3.4 Transition Graphs and Fixed Points [124] 3.5 Basins of Attraction [129] Challenge 3: Sharkovskii’s Theorem [135] Exercises [140] Lab Visit 3: Periodicity and Chaos in a Chemical Reaction [143] 4 FRACTALS [149] 4.1 Cantor Sets [150] 4.2 Probabilistic Constructions of Fractals [156] 4-3 Fractals from Deterministic Systems [161] 4.4 Fractal Basin Boundaries [164] 4.5 Fractal Dimension [172] 4-6 Computing the Box-Counting Dimension [177] 4.7 Correlation Dimension [180] Challenge 4: Fractal Basin Boundaries and the Uncertainty Exponent [183] Exercises [186] Lab Visit 4: Fractal Dimension in Experiments [188] 5 CHAOS IN TWO-DIMENSIONAL MAPS [193] 5.1 Lyapunov Exponents [194] 5.2 Numerical Calculation of Lyapunov Exponents [199] 5.3 Lyapunov Dimension [203] 5.4 A Two-Dimensional Fixed-Point Theorem [207] 5.5 Markov Partitions [212] 5.6 The Horseshoe Map [216] Challenge 5: Computer Calculations and Shadowing [222] Exercises [226] Lab Visit 5: Chaos in Simple Mechanical Devices [228] 6 CHAOTIC ATTRACTORS [231] 6.1 Forward Limit Sets [233] 6.2 Chaotic Attractors [238] 6.3 Chaotic Attractors of Expanding Interval Maps [245] 6.4 Measure [249] 6.5 Natural Measure [253] 6.6 Invariant Measure for One-Dimensional Maps [256] Challenge 6: Invariant Measure for the Logistic Map [264] Exercises [266] Lab Visit 6: Fractal Scum [267] 7 DIFFERENTIAL EQUATIONS [273] 7.1 One-Dimensional Linear Differential Equations [275] 7.2 One-Dimensional Nonlinear Differential Equations [278] 7.3 Linear Differential Equations in More than One Dimension [284] 7.4 Nonlinear Systems [294] 7.5 Motion in a Potential Field [300] 7.6 Lyapunov Functions [304] 7.7 Lotka-Volterra Models [309] Challenge 7: A Limit Cycle in the Van der Pol System [316] Exercises [321] Lab Visit 7: Fly vs. Fly [325] 8 PERIODIC ORBITS AND LIMIT SETS [329] 8.1 Limit Sets for Planar Differential Equations [331] 8.2 Properties of w-Limit Sets [337] 8.3 Proof of the Poincare-Bendixson Theorem [341] Challenge 8: Two Incommensurate Frequencies Form a Torus [350] Exercises [353] Lab Visit 8: Steady States and Periodicity in a Squid Neuron [355] 9 CHAOS IN DIFFERENTIAL EQUATIONS [359] 9.1 The Lorenz Attractor [359] 9.2 Stability in the Large, Instability in the Small [366] 9.3 The Rössler Attractor [370] 9.4 Chua’s Circuit [375] 9.5 Forced Oscillators [376] 9.6 Lyapunov Exponents in Flows [379] CHALLENGE 9: SYNCHRONIZATION OF CHAOTIC ORBITS [387] Exercises [393] Lab Visit 9: Lasers in Synchronization [394] 10 STABLE MANIFOLDS AND CRISES [399] 10.1 The Stable Manifold Theorem [401] 10.2 Homoclinic and Heteroclinic Points [409] 10.3 Crises [413] 10.4 Proof of the Stable Manifold Theorem [422] 10.5 Stable and Unstable Manifolds for Higher Dimensional Maps [430] Challenge 10: The Lakes of Wada [432] Exercises [440] Lab Visit 10: The Leaky Faucet: Minor Irritation or Crisis? [441] 11 BIFURCATIONS [447] 11.1 Saddle-Node and Period-Doubling Bifurcations [448] 11.2 Bifurcation Diagrams [453] 11.3 Continuability [460] 11.4 Bifurcations of One-Dimensional Maps [464] 11.5 Bifurcations in Plane Maps: Area-Contracting Case [468] 11.6 Bifurcations in Plane Maps: Area-Preserving Case [471] 11.7 Bifurcations in Differential Equations [478] 1L8 Hopf Bifurcations [483] Challenge 11: Hamiltonian Systems and the Lyapunov Center Theorem [491] Exercises [494] Lab Visit 11: iron + Sulfuric Acid —> Hopf Bifurcation [496] 12 CASCADES [499] 12.1 Cascades and 4-669201609 [500] 12.2 Schematic Bifurcation Diagrams [504] 12.3 Generic Bifurcations [510] 12.4 The Cascade Theorem [518] Challenge 12: Universality in Bifurcation Diagrams 525 Exercises [531] Lab Visit 12: Experimental Cascades [532] 13 STATE RECONSTRUCTION FROM DATA [537] 13.1 Delay Plots from Time Series [537] 13.2 Delay Coordinates [541] 13.3 Embedology [545] Challenge 13: Box-Counting Dimension and Intersection [553] A MATRIX ALGEBRA [557] A.l Eigenvalues and Eigenvectors [557] A.2 Coordinate Changes [561] A. 3 Matrix Tunes Circle Equals Ellipse [563] B COMPUTER SOLUTION OF ODES [567] B. 1 ODE Solvers [568] B.2 Error in Numerical Integration [570] B.3 Adaptive Step-Size Methods [574] ANSWERS AND HINTS TO SELECTED EXERCISES [577] BIBLIOGRAPHY [587] INDEX [595]
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Includes bibliographical references (p. 587-594) and index.
Contents --
INTRODUCTION v --
I ONE-DIMENSIONAL MAPS I --
1.1 One-Dimensional Maps [2] --
1.2 Cobweb Plot: Graphical Representation of an Orbit [5] --
1.3 Stability of Fixed Points [9] --
1.4 Periodic Points [13] --
1.5 The Family of Logistic Maps [17] --
1.6 The Logistic Map G (x) = 4x( 1 — x) [22] --
1.7 Sensitive Dependence on Initial Conditions [25] --
1.8 Itineraries [27] --
Challenge 1: Period Three Implies Chaos [32] --
Exercises [35] --
Lab Visit 1: Boom, Bust, and Chaos in the Beetle Census [39] --
2 TWO-DIMENSIONAL MAPS [43] --
2.1 Mathematical Models [44] --
2.2 Sinks, Sources, and Saddles [58] --
2.3 Linear Maps [62] --
2.4 Coordinate Changes [67] --
2.5 Nonlinear Maps and the Jacobian Matrix [68] --
2.6 Stable and Unstable Manifolds [78] --
2.7 Matrix Times Circle Equals Ellipse [87] --
Challenge 2: Counting the Periodic Orbits of Linear Maps on a Torus [92] --
Exercises [98] --
Lab Visit 2: Is the Solar System Stable? [99] --
3 CHAOS [105] --
3.1 Lyapunov Exponents [106] --
3.2 Chaotic Orbits [109] --
3.3 Conjugacy and the Logistic Map [114] --
3.4 Transition Graphs and Fixed Points [124] --
3.5 Basins of Attraction [129] --
Challenge 3: Sharkovskii’s Theorem [135] --
Exercises [140] --
Lab Visit 3: Periodicity and Chaos in a Chemical Reaction [143] --
4 FRACTALS [149] --
4.1 Cantor Sets [150] --
4.2 Probabilistic Constructions of Fractals [156] --
4-3 Fractals from Deterministic Systems [161] --
4.4 Fractal Basin Boundaries [164] --
4.5 Fractal Dimension [172] --
4-6 Computing the Box-Counting Dimension [177] --
4.7 Correlation Dimension [180] --
Challenge 4: Fractal Basin Boundaries and the Uncertainty Exponent [183] --
Exercises [186] --
Lab Visit 4: Fractal Dimension in Experiments [188] --
5 CHAOS IN TWO-DIMENSIONAL MAPS [193] --
5.1 Lyapunov Exponents [194] --
5.2 Numerical Calculation of Lyapunov Exponents [199] --
5.3 Lyapunov Dimension [203] --
5.4 A Two-Dimensional Fixed-Point Theorem [207] --
5.5 Markov Partitions [212] --
5.6 The Horseshoe Map [216] --
Challenge 5: Computer Calculations and Shadowing [222] --
Exercises [226] --
Lab Visit 5: Chaos in Simple Mechanical Devices [228] --
6 CHAOTIC ATTRACTORS [231] --
6.1 Forward Limit Sets [233] --
6.2 Chaotic Attractors [238] --
6.3 Chaotic Attractors of Expanding Interval Maps [245] --
6.4 Measure [249] --
6.5 Natural Measure [253] --
6.6 Invariant Measure for One-Dimensional Maps [256] --
Challenge 6: Invariant Measure for the Logistic Map [264] --
Exercises [266] --
Lab Visit 6: Fractal Scum [267] --
7 DIFFERENTIAL EQUATIONS [273] --
7.1 One-Dimensional Linear Differential Equations [275] --
7.2 One-Dimensional Nonlinear Differential Equations [278] --
7.3 Linear Differential Equations in More than One Dimension [284] --
7.4 Nonlinear Systems [294] --
7.5 Motion in a Potential Field [300] --
7.6 Lyapunov Functions [304] --
7.7 Lotka-Volterra Models [309] --
Challenge 7: A Limit Cycle in the Van der Pol System [316] --
Exercises [321] --
Lab Visit 7: Fly vs. Fly [325] --
8 PERIODIC ORBITS AND LIMIT SETS [329] --
8.1 Limit Sets for Planar Differential Equations [331] --
8.2 Properties of w-Limit Sets [337] --
8.3 Proof of the Poincare-Bendixson Theorem [341] --
Challenge 8: Two Incommensurate Frequencies Form a Torus [350] --
Exercises [353] --
Lab Visit 8: Steady States and Periodicity in a Squid Neuron [355] --
9 CHAOS IN DIFFERENTIAL EQUATIONS [359] --
9.1 The Lorenz Attractor [359] --
9.2 Stability in the Large, Instability in the Small [366] --
9.3 The Rössler Attractor [370] --
9.4 Chua’s Circuit [375] --
9.5 Forced Oscillators [376] --
9.6 Lyapunov Exponents in Flows [379] --
CHALLENGE 9: SYNCHRONIZATION OF CHAOTIC ORBITS [387] --
Exercises [393] --
Lab Visit 9: Lasers in Synchronization [394] --
10 STABLE MANIFOLDS AND CRISES [399] --
10.1 The Stable Manifold Theorem [401] --
10.2 Homoclinic and Heteroclinic Points [409] --
10.3 Crises [413] --
10.4 Proof of the Stable Manifold Theorem [422] --
10.5 Stable and Unstable Manifolds for Higher Dimensional Maps [430] --
Challenge 10: The Lakes of Wada [432] --
Exercises [440] --
Lab Visit 10: The Leaky Faucet: Minor Irritation or Crisis? [441] --
11 BIFURCATIONS [447] --
11.1 Saddle-Node and Period-Doubling Bifurcations [448] --
11.2 Bifurcation Diagrams [453] --
11.3 Continuability [460] --
11.4 Bifurcations of One-Dimensional Maps [464] --
11.5 Bifurcations in Plane Maps: Area-Contracting Case [468] --
11.6 Bifurcations in Plane Maps: Area-Preserving Case [471] --
11.7 Bifurcations in Differential Equations [478] --
1L8 Hopf Bifurcations [483] --
Challenge 11: Hamiltonian Systems and the Lyapunov Center Theorem [491] --
Exercises [494] --
Lab Visit 11: iron + Sulfuric Acid —> Hopf Bifurcation [496] --
12 CASCADES [499] --
12.1 Cascades and 4-669201609 [500] --
12.2 Schematic Bifurcation Diagrams [504] --
12.3 Generic Bifurcations [510] --
12.4 The Cascade Theorem [518] --
Challenge 12: Universality in Bifurcation Diagrams 525 Exercises [531] --
Lab Visit 12: Experimental Cascades [532] --
13 STATE RECONSTRUCTION FROM DATA [537] --
13.1 Delay Plots from Time Series [537] --
13.2 Delay Coordinates [541] --
13.3 Embedology [545] --
Challenge 13: Box-Counting Dimension --
and Intersection [553] --
A MATRIX ALGEBRA [557] --
A.l Eigenvalues and Eigenvectors [557] --
A.2 Coordinate Changes [561] --
A. 3 Matrix Tunes Circle Equals Ellipse [563] --
B COMPUTER SOLUTION OF ODES [567] --
B. 1 ODE Solvers [568] --
B.2 Error in Numerical Integration [570] --
B.3 Adaptive Step-Size Methods [574] --
ANSWERS AND HINTS TO SELECTED EXERCISES [577] --
BIBLIOGRAPHY [587] --
INDEX [595] --
MR,
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