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Chaos in nature

Author: Christophe Letellier
Publisher: New Jersey : World Scientific, [2013]
Series: World Scientific series on nonlinear science., Series A,, Monographs and treatises ;, v. 81.
Edition/Format:   Print book : English
Summary:

Chaos theory deals with the description of motion (in a general sense) which cannot be predicted in the long term although produced by deterministic system, as well exemplified by meteorological  Read more...

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Document Type: Book
All Authors / Contributors: Christophe Letellier
ISBN: 9789814374422 9814374423
OCLC Number: 840187098
Description: xiv, 378 pages : illustrations ; 24 cm.
Contents: From Celestial Mechanics to Chaos 1 --
1 The Laws of Dynamics 3 --
1.1 Kepler's Empirical Laws 3 --
1.2 The Law of Gravitation 6 --
1.3 Theory of the Moon 11 --
2 The Three Body Problem 13 --
2.1 Imperfections in Newton's Theory 13 --
2.2 Challenges to the Law of Gravitation 14 --
2.3 Problem of the Convergence of Series 19 --
3 Simplification of the Three Body Problem 23 --
3.1 Simplification of the Geometry 23 --
3.2 Simplification of the General Equations 25 --
3.3 The First Exact Solutions 29 --
4 The Success of Celestial Mechanics 35 --
4.1 Perturbation Theory 35 --
4.2 The Theory of Jupiter and Saturn 38 --
4.3 The Theory of the Moon 40 --
4.4 Laplacian Determinism 41 --
4.5 The Discovery of Neptune 44 --
4.6 The Development of Perturbation Theory 48 --
5 Birth of the Global Approach 51 --
5.1 The Restricted Three-Body Problem 51 --
5.2 A Qualitative Approach 55 --
5.3 Studies of Sets of Solutions 56 --
5.4 Dynamical Systems 58 --
5.5 The Ideal Pendulum 59 --
5.6 The Poincaré-Bendixon Theorem 63 --
5.7 Doubly Asymptotic Orbits 65 --
5.8 Deterministic but Unpredictable 72 --
6 The Stability of the Solar System 73 --
6.1 The Problem of Small Devisors 74 --
6.2 The KAM Theorem 76 --
6.3 A Model for the KAM Theorem 79 --
6.4 Numerical Approach 83 --
Chaos in Nature: Properties and Examples 87 --
1 Periodic and Chaotic Oscillators 89 --
1.1 Oscillators and Degrees of Freedom 90 --
1.2 Damped Pendulum 93 --
1.3 Linear System of Two Oscillators 94 --
1.4 Nonlinear System of Two Oscillators 96 --
2 From Mathematics to Electronic Circuits 101 --
2.1 The Early Self-Oscillating Systems 103 --
2.1.1 The series dynamo machine 103 --
2.1.2 The musical arc 103 --
2.1.3 From vacuum tubes to oscillating valves 107 --
2.1.4 From the audion to the multivibrator 111 --
2.2 The First Dynamical Studies of Oscillators 113 --
2.2.1 Poincarée's equation for the musical arc 113 --
2.2.2 Janet's equation for series dynamo machine 115 --
2.2.3 Blondel's equation for the triode 116 --
2.2.4 The van der Pol Equation 116 --
2.2.5 Some equations for the multivibrator and beyond 120 --
2.3 Relaxation Oscillations 126 --
2.3.1 First insights from the German school 126 --
2.3.2 Van der Pol's contribution 128 --
2.3.3 Relaxation oscillations in the real world 132 --
2.4 The First Computer Calculations 136 --
2.5 First Chaotic Attractors in Electronic Circuits 142 --
2.6 A Chaotic Thermionic Diode 157 --
3 From Meteorology to Chaos: The Second Wave 163 --
3.1 Prediction in Meteorology 163 --
3.2 The Lorenz System 167 --
3.2.1 Phase space 168 --
3.2.2 The stability of periodic solutions 170 --
3.2.3 Numerical integration and application of linear theory 171 --
3.2.4 Topological analysis 172 --
3.2.5 First-return map to maxima 173 --
3.3 Sensitivity to Initial Conditions 175 --
3.4 Turbulence, A periodic Solutions, and Chaos 178 --
3.5 Hydrodynamics and the Lorenz Attractor 181 --
3.6 Laser Dynamics and the Lorenz System 182 --
4 The Architecture of Chaotic Attractors 189 --
4.1 The Rössler System 189 --
4.1.1 A brief biography 189 --
4.1.2 Rössler's main influences 195 --
4.1.3 A chaotic chemical reaction 200 --
4.1.4 The Rössler system 206 --
4.1.5 A forgotten topological analysis 209 --
4.2 Poincaré Section 215 --
4.3 Symbolic Dynamics 216 --
4.4 Topological Characterization 220 --
4.5 A Simple Model for the Poincaré Map 222 --
4.6 Different Topologies for Chaos 229 --
5 Chemical Reactions 235 --
5.1 The Earliest Experiments 235 --
5.2 Chaos in an Experimental BZ-Reaction 240 --
5.3 Chaotic Copper Electrodissolution 246 --
6 Population Evolution 255 --
6.1 Theories of Malthus and Verhulst 255 --
6.2 A Model with Two Species 260 --
6.3 Models with Three Species 264 --
6.4 Observational Evidence 270 --
7 Chaos in Biology and Biomedicine 277 --
7.1 Glycolysis Oscillations 277 --
7.2 Fluctuations in Hematopoiesis 282 --
7.3 Cardiac Arrhythmias 284 --
7.3.1 The beginnings of electrophysiology 285 --
7.3.2 The heart --
An electric machine 290 --
7.3.3 Electrocardiograms and arrhythmias 293 --
7.3.4 Analysis of some heart rate variability 297 --
7.4 Patient Breathing with a Noninvasive Mechanical Ventilation 308 --
7.4.1 Early techniques for mechanical ventilation 308 --
7.4.2 Breathing variability under mechanical ventilation 312 --
7.5 Conclusion 322 --
8 Chaotically Variable Stars 323 --
8.1 The First Observations 324 --
8.2 The First Chaotic Models 330 --
8.3 Solar Activity 339 --
8.4 Chaotic Models of Solar Activity 350 --
9 Epilogue 361 --
9.1 The Fourth Dimension 361 --
9.2 A Weakly Dissipative System 363 --
9.3 Hyperchaotic Behavior 364 --
9.4 Toroidal Chaos 365 --
9.5 Simple Models and Complex Behaviors 366.
Series Title: World Scientific series on nonlinear science., Series A,, Monographs and treatises ;, v. 81.
Responsibility: Christophe Letellier.

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