| Description |
1 online resource (xxi, 323 pages) : illustrations |
| Series |
Cambridge nonlinear science series ; 2 |
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Cambridge nonlinear science series ; 2.
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| Contents |
Cover; Half Title; Title Page; Copyright; Dedication; Contents; Preface; Introduction; 1 Flows in phase space; 1.1 Determinism, phase flows, and Liouville's theorem; 1.2 Equilibria, linear stability, and limit cycles; 1.3 Change of stability (bifurcations); 1.4 Periodically driven systems and stroboscopic maps; 1.5 Continuous groups of transformations as phase space flows; 2 Introduction to deterministic chaos; 2.1 The Lorenz model, the Lorenz plot, and the binary tent map; 2.2 Local exponential instability of nearby orbits: the positiveLiapunov exponent |
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2.3 The Frobenius-Peron equation (invariant densities)2.4 Simple examples of fully developed chaos for maps of theinterval; 2.5 Maps that are conjugate under differentiable coordinatetransformations; 2.6 Computation of nonperiodic chaotic orbits at fullydeveloped chaos; 2.7 Is the idea of randomness necessary in natural science?; 3 Conservative dynamical systems; 3.1 Integrable conservative systems: symmetry, invariance, conservation laws, and motion on invariant tori in phase space; 3.2 The Hénon-Heiles model: evidence for bifurcations from integrable to chaotic behavior |
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3.3 Perturbed twist maps: nearly integrable conservativesystems3.4 Mixing and ergodicity: the approach to statisticalequilibrium; 3.5 The bakers' transformation; 3.6 Computation of chaotic orbits for an area-preserving map; Appendix 3.A Generating functions for canonicaltransformations; Appendix 3.B Systems in involution; 4 Fractals and fragmentation in phase space; 4.1 Introduction to fractals; 4.2 Geometrically selfsimilar fractals; 4.3 The dissipative bakers' transformation: a model 'strange' attractor; 4.4 The symmetric tent map: a model 'strange' repeller |
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4.5 The devil's staircase: arithmetic on the Cantor set4.6 Generalized dimensions and the coarsegraining of phasespace; 4.7 Computation of chaotic orbits on a fractal; 5 The way to chaos by instability of quasiperiodic orbits; 5.1 From limit cycles to tori to chaos; 5.2 Periodically driven systems and circle maps; 5.3 Arnol'd tongues and the devil's staircase; 5.4 Scaling laws and renormalization group equations; 5.5 The Farey tree; 6 The way to chaos by period doubling; 6.1 Universality at transitions to chaos; 6.2 Instability of periodic orbits by period doubling |
| Summary |
This book develops deterministic chaos and fractals from the standpoint of iterated maps, but the emphasis makes it very different from all other books in the field. It provides the reader with an introduction to more recent developments, such as weak universality, multifractals, and shadowing, as well as to older subjects like universal critical exponents, devil's staircases and the Farey tree. The author uses a fully discrete method, a 'theoretical computer arithmetic', because finite (but not fixed) precision cannot be avoided in computation or experiment. This leads to a more general formulation in terms of symbolic dynamics and to the idea of weak universality. The connection is made with Turing's ideas of computable numbers and it is explained why the continuum approach leads to predictions that are not necessarily realized in computation or in nature, whereas the discrete approach yields all possible histograms that can be observed or computed |
| Bibliography |
Includes bibliographical references (pages 309-317) and index |
| Notes |
Print version record |
| Subject |
Deterministic chaos.
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Algorithms.
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Mappings (Mathematics)
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Fractals.
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Mathematical physics.
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Algorithms
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algorithms.
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fractals.
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MATHEMATICS -- Calculus.
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MATHEMATICS -- Mathematical Analysis.
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Algorithms
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Deterministic chaos
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Fractals
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Mappings (Mathematics)
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Mathematical physics
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Chaostheorie
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Fraktal
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Chaos, (théorie des systèmes)
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Algorithmes.
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Fractales.
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| Form |
Electronic book
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| ISBN |
9780511564154 |
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0511564155 |
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9781461951742 |
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1461951747 |
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9781107390157 |
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110739015X |
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0521467470 |
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9780521467476 |
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9781107387300 |
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1107387302 |
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