| Description |
1 online resource |
| Series |
Studies in nonlinearity |
|
Studies in nonlinearity.
|
| Contents |
Cover; Half Title; Title Page; Copyright Page; Dedication; Preface; Table of Contents; 1: Introduction; 1.1 Dynamical models; 1.2 Celestial mechanics; 1.3 Lorenz: the end of weather prediction?; 1.4 Complex behaviour of simple systems; 2: Orbits of one-dimensional systems; 2.1 Discrete dynamical systems; 2.2 Fixed points and stability; 2.3 Some orbits of the tent map; 2.4 Period doubling of the logistic map; 2.5 Periodic orbits and compositions; 2.6 The fully chaotic tent map; 2.7 Numerical versus exact orbits; 2.8 Fourier analysis of an orbit; 2.9 Lyapunov exponent of an orbit |
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2.10 Chaotic orbits2.11 Ergodic orbits; 3: Bifurcations in one-dimensional systems; 3.1 Bifurcation diagrams; 3.2 Final state diagrams; 3.3 Period doubling mechanism; 3.4 Period doubling cascades; 3.5 Feigenbaum's universal constants; 3.6 Tangent bifurcations; 3.7 Intermittent behaviour; 3.8 Unstable orbits and crises; 3.9 Transcritical and pitchfork bifurcations; 3.10 Theory of Feigenbaum scaling; 4: Two-dimensional systems; 4.1 The Hénon map; 4.2 Fixed points; 4.3 Area contraction; 4.4 Stability of fixed points; 4.5 Periodic orbits; 4.6 Lyapunov exponents; 4.7 Basin boundaries |
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4.8 Linear feedback and control4.9 Tangent bifurcations and zero curves; 4.10 Producing the strange attractor; 5: Fractals; 5.1 The Hénon attractor; 5.2 The Cantor Set; 5.3 Fractal bifurcation diagrams; 5.4 Capacity dimension; 5.5 Capacity dimension of the Hénon attractor; 5.6 Self-similar fractals; 5.7 Lyapunov dimension of two-dimensional maps; 5.8 The Rössler attractor; 5.9 The Feigenbaum attractor; 6: Non-linear oscillations; 6.1 The driven non-linear pendulum; 6.2 Phase Plane; 6.3 Poincaré sections; 6.4 Lyapunov exponents; A: Chaos for Java Software; A.1 Installation |
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A.2 General featuresA. 3 Menus; A.4 BIFURCATION DIAGRAMS; A.5 FOURIER ANALYSIS; A.6 GRAPHICAL ANALYSIS; A.7 lTERATE(1D); A.8 lTERATE(2D); A.9 LYAPUNOV EXPONENTS; A.10 ODE ORBITS; A.11 POINCARÉ SECTIONS; A.12 RETURN MAPS; B: Discrete Fourier Transform; B.1 Complex roots of unity; B.2 Discrete orthogonality; B.3 Fourier amplitudes; B.4 Using real and imaginary parts; B.5 Fast Fourier Transform; C: Variational equations; C.1 Derivation and meaning; C.2 The area contracting property; C.3 Three-dimensional case; D: List of Maps and Differential Equations; D.1 One-dimensional maps |
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D.2 Two-dimensional mapsD. 3 Differential equations; Bibliography; Index |
| Summary |
This book presents elements of the theory of chaos in dynamical systems in a framework of theoretical understanding coupled with numerical and graphical experimentation. The theory is developed using only elementary calculus and algebra, and includes dynamics of one-and two-dimensional maps, periodic orbits, stability and its quantification, chaotic behavior, and bifurcation theory of one-dimensional systems. There is an introduction to the theory of fractals, with an emphasis on the importance of scaling, and a concluding chapter on ordinary differential equations. The accompanying software, written in Java, is available online (see link below) The program enables students to carry out their own quantitative experiments on a variety of nonlinear systems, including the analysis of fixed points of compositions of maps, calculation of Fourier spectra and Lyapunov exponents, and box counting for two-dimensional maps. It also provides for visualizing orbits, final state and bifurcation diagrams, Fourier spectra and Lyapunov exponents, basins of attractions, three-dimensional orbits, Poincar?ections, and return maps. Please visit http://www.maths.anu.edu.au/b̃riand/chaos/ for the integrated cross-platform software |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Online resource; title from PDF title page (EBSCO, viewed May 22, 2018) |
| Subject |
Chaotic behavior in systems.
|
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Differentiable dynamical systems.
|
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SCIENCE -- System Theory.
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TECHNOLOGY & ENGINEERING -- Operations Research.
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Chaotic behavior in systems
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Differentiable dynamical systems
|
| Form |
Electronic book
|
| ISBN |
9780786745616 |
|
0786745614 |
|
9780429971419 |
|
0429971419 |
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