
Chaos in Nature.
Title:
Chaos in Nature.
Author:
Letellier, Christophe.
ISBN:
9789814374439
Personal Author:
Physical Description:
1 online resource (393 pages)
Series:
World Scientific Series on Nonlinear Science: Series A
Contents:
Contents -- Foreword by Otto E. Rossler -- Foreword by Robert Gilmore -- From Celestial Mechanics to Chaos -- 1. The Laws of Dynamics -- 1.1 Kepler's Empirical Laws -- 1.2 The Law of Gravitation -- 1.3 Theory of the Moon -- 2. The Three Body Problem -- 2.1 Imperfections in Newton's Theory -- 2.2 Challenges to the Law of Gravitation -- 2.3 Problem of the Convergence of Series -- 3. Simplification of the Three Body Problem -- 3.1 Simplification of the Geometry -- 3.2 Simplification of the General Equations -- 3.3 The First Exact Solutions -- 4. The Success of Celestial Mechanics -- 4.1 Perturbation Theory -- 4.2 The Theory of Jupiter and Saturn -- 4.3 The Theory of the Moon -- 4.4 Laplacian Determinism -- 4.5 The Discovery of Neptune -- 4.6 The Development of Perturbation Theory -- 5. Birth of the Global Approach -- 5.1 The Restricted Three-Body Problem -- 5.2 A Qualitative Approach -- 5.3 Studies of Sets of Solutions -- 5.4 Dynamical Systems -- 5.5 The Ideal Pendulum -- 5.6 The Poincare-Bendixon Theorem -- 5.7 Doubly Asymptotic Orbits -- 5.8 Deterministic but Unpredictable -- 6. The Stability of the Solar System -- 6.1 The Problem of Small Devisors -- 6.2 The KAM Theorem -- 6.3 A Model for the KAM Theorem -- 6.4 Numerical Approach -- Chaos in Nature: Properties and Examples -- 1. Periodic and Chaotic Oscillators -- 1.1 Oscillators and Degrees of Freedom -- 1.2 Damped Pendulum -- 1.3 Linear System of Two Oscillators -- 1.4 Nonlinear System of Two Oscillators -- 2. From Mathematics to Electronic Circuits -- 2.1 The Early Self-Oscillating Systems -- 2.1.1 The series dynamo machine -- 2.1.2 The musical arc -- 2.1.3 From vacuum tubes to oscillating valves -- 2.1.4 From the audion to the multivibrator -- 2.2 The First Dynamical Studies of Oscillators -- 2.2.1 Poincare's equation for the musical arc -- 2.2.2 Janet's equation for series dynamo machine.
2.2.3 Blondel's equation for the triode -- 2.2.4 The van der Pol Equation -- 2.2.5 Some equations for the multivibrator and beyond -- 2.3 Relaxation Oscillations -- 2.3.1 First insights from the German school -- 2.3.2 Van der Pol's contribution -- 2.3.3 Relaxation oscillations in the real world -- 2.4 The First Computer Calculations -- 2.5 First Chaotic Attractors in Electronic Circuits -- 2.6 A Chaotic Thermionic Diode -- 3. From Meteorology to Chaos: The Second Wave -- 3.1 Prediction in Meteorology -- 3.2 The Lorenz System -- 3.2.1 Phase space -- 3.2.2 The stability of periodic solutions -- 3.2.3 Numerical integration and application of linear theory -- 3.2.4 Topological analysis -- 3.2.5 First-return map to maxima -- 3.3 Sensitivity to Initial Conditions -- 3.4 Turbulence, Aperiodic Solutions, and Chaos -- 3.5 Hydrodynamics and the Lorenz Attractor -- 3.6 Laser Dynamics and the Lorenz System -- 4. The Architecture of Chaotic Attractors -- 4.1 The Rossler System -- 4.1.1 A brief biography -- 4.1.2 Rossler's main influences -- 4.1.3 A chaotic chemical reaction -- 4.1.4 The Rossler system -- 4.1.5 A forgotten topological analysis -- 4.2 Poincare Section -- 4.3 Symbolic Dynamics -- 4.4 Topological Characterization -- 4.5 A Simple Model for the Poincare Map -- 4.6 Different Topologies for Chaos -- 5. Chemical Reactions -- 5.1 The Earliest Experiments -- 5.2 Chaos in an Experimental BZ-Reaction -- 5.3 Chaotic Copper Electrodissolution -- 6. Population Evolution -- 6.1 Theories of Malthus and Verhulst -- 6.2 A Model with Two Species -- 6.3 Models with Three Species -- 6.4 Observational Evidence -- 7. Chaos in Biology and Biomedicine -- 7.1 Glycolysis Oscillations -- 7.2 Fluctuations in Hematopoiesis -- 7.3 Cardiac Arrhythmias -- 7.3.1 The beginnings of electrophysiology -- 7.3.2 The heart - An electric machine -- 7.3.3 Electrocardiograms and arrhythmias.
7.3.4 Analysis of some heart rate variability -- 7.4 Patient Breathing with a Noninvasive Mechanical Ventilation -- 7.4.1 Early techniques for mechanical ventilation -- 7.4.2 Breathing variability under mechanical ventilation -- 7.5 Conclusion -- 8. Chaotically Variable Stars -- 8.1 The First Observations -- 8.2 The First Chaotic Models -- 8.3 Solar Activity -- 8.4 Chaotic Models of Solar Activity -- 9. Epilogue -- 9.1 The Fourth Dimension -- 9.2 A Weakly Dissipative System -- 9.3 Hyperchaotic Behavior -- 9.4 Toroidal Chaos -- 9.5 Simple Models and Complex Behaviors -- General Index -- Author Index.
Abstract:
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 phenomena. It directly comes from the Lunar theory - a three-body problem - and the difficulty encountered by astronomers to accurately predict the long-term evolution of the Moon using "Newtonian" mechanics. Henri Poincaré's deep intuitions were at the origin of chaos theory. They also led the meteorologist Edward Lorenz to draw the first chaotic attractor ever published. But the main idea consists of plotting a curve representative of the system evolution rather than finding an analytical solution as commonly done in classical mechanics. Such a novel approach allows the description of population interactions and the solar activity as well. Using the original sources, the book draws on the history of the concepts underlying chaos theory from the 17th century to the last decade, and by various examples, show how general is this theory in a wide range of applications: meteorology, chemistry, populations, astrophysics, biomedicine, etc.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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