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BL
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| Record Number
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878515
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| Main Entry
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Scheinerman, Edward R.
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| Title & Author
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Invitation to dynamical systems /\ Edward R. Scheinerman.
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| Edition Statement
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Dover ed.
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| Publication Statement
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Mineola, N.Y. :: Dover Publications,, 2012.
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| Series Statement
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Dover Books on Mathematics
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| Page. NO
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1 online resource (xvii, 373 pages) :: illustrations
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| ISBN
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0486275329
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: 9780486275321
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0486485943
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9780486485942
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| Notes
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Originally published: Upper Saddle River, N.J. : Prentice-Hall, 1996.
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| Bibliographies/Indexes
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Includes bibliographical references (pages 369-370) and index.
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| Contents
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Cover Page; Title Page; Copyright Page; Dedication; Contents; Preface; 1 Introduction; 1.1 What is a dynamical system?; 1.1.1 State vectors; 1.1.2 The next instant: discrete time; 1.1.3 The next instant: continuous time; 1.1.4 Summary; Problems; 1.2 Examples; 1.2.1 Mass and spring; 1.2.2 RLC circuits; 1.2.3 Pendulum; 1.2.4 Your bank account; 1.2.5 Economic growth; 1.2.6 Pushing buttons on your calculator; 1.2.7 Microbes; 1.2.8 Predator and prey; 1.2.9 Newton's Method; 1.2.10 Euler's method; 1.2.11 "Random" number generation; Problems; 1.3 What we want; what we can get; 2 Linear Systems.
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2.1 One dimension2.1.1 Discrete time; 2.1.2 Continuous time; 2.1.3 Summary; Problems; 2.2 Two (and more) dimensions; 2.2.1 Discrete time; 2.2.2 Continuous time; 2.2.3 The nondiagonalizable case*; Problems; 2.3 Examplification: Markov chains; 2.3.1 Introduction; 2.3.2 Markov chains as linear systems; 2.3.3 The long term; Problems; 3 Nonlinear Systems 1: Fixed Points; 3.1 Fixed points; 3.1.1 What is a fixed point?; 3.1.2 Finding fixed points; 3.1.3 Stability; Problems; 3.2 Linearization; 3.2.1 One dimension; 3.2.2 Two and more dimensions; Problems; 3.3 Lyapunov functions.
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3.3.1 Linearization can fail3.3.2 Energy; 3.3.3 Lyapunov's method; 3.3.4 Gradient systems; Problems; 3.4 Examplification: Iterative methods for solving equations; Problems; 4 Nonlinear Systems 2: Periodicity and Chaos; 4.1 Continuous time; 4.1.1 One dimension: no periodicity; 4.1.2 Two dimensions: the Poincaré-Bendixson theorem; 4.1.3 The Hopf bifurcation*; 4.1.4 Higher dimensions: the Lorenz system and chaos; Problems; 4.2 Discrete time; 4.2.1 Periodicity; 4.2.2 Stability of periodic points; 4.2.3 Bifurcation; 4.2.4 Sarkovskii's theorem*; 4.2.5 Chaos and symbolic dynamics; Problems.
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4.3 Examplification: Riffle shuffles and the shift map4.3.1 Riffle shuffles; 4.3.2 The shift map; 4.3.3 Shifting and shuffling; 4.3.4 Shuffling again and again; Problems; 5 Fractals; 5.1 Cantor's set; 5.1.1 Symbolic representation of Cantor's set; 5.1.2 Cantor's set in conventional notation; 5.1.3 The link between the two representations; 5.1.4 Topological properties of the Cantor set; 5.1.5 In what sense a fractal?; Problems; 5.2 Biting out the middle in the plane; 5.2.1 Sierpinski's triangle; 5.2.2 Koch's snowflake; Problems; 5.3 Contraction mapping theorems; 5.3.1 Contraction maps.
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5.3.2 Contraction mapping theorem on the real line5.3.3 Contraction mapping in higher dimensions; 5.3.4 Contractive affine maps: the spectral norm*; 5.3.5 Other metric spaces; 5.3.6 Compact sets and Hausdorff distance; Problems; 5.4 Iterated function systems; 5.4.1 From point maps to set maps; 5.4.2 The union of set maps; 5.4.3 Examples revisited; 5.4.4 IFSs defined; 5.4.5 Working backward; Problems; 5.5 Algorithms for drawing fractals; 5.5.1 A deterministic algorithm; 5.5.2 Dancing on fractals; 5.5.3 A randomized algorithm; Problems; 5.6 Fractal dimension; 5.6.1 Covering with balls.
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| Abstract
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This text is designed for those who wish to study mathematics beyond linear algebra but are not ready for abstract material. Rather than a theorem-proof-corollary-remark style of exposition, it stresses geometry, intuition, and dynamical systems. An appendix explains how to write MATLAB, Mathematica, and C programs to compute dynamical systems. 1996 edition.
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| Subject
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Differentiable dynamical systems.
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| Subject
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Differentiable dynamical systems.
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| Subject
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MATHEMATICS-- Geometry-- Differential.
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| Dewey Classification
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003/.85
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| LC Classification
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QA614.8.S34 2012eb
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