Author | Alligood, Kathleen T. author |
---|---|
Title | Chaos [electronic resource] : An Introduction to Dynamical Systems / by Kathleen T. Alligood, Tim D. Sauer, James A. Yorke |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1997 |
Connect to | http://dx.doi.org/10.1007/978-3-642-59281-2 |
Descript | XVII, 603 p. online resource |
1 One-Dimensional Maps -- 1.1 One-Dimensional Maps -- 1.2 Cobweb Plot: Graphical Representation of an Orbit -- 1.3 Stability of Fixed Points -- 1.4 Periodic Points -- 1.5 The Family of Logistic Maps -- 1.6 The Logistic Map G(x) = 4x(l ? x) -- 1.7 Sensitive Dependence on Initial Conditions -- 1.8 Itineraries -- 2 Two-Dimensional Maps -- 2.1 Mathematical Models -- 2.2 Sinks, Sources, and Saddles -- 2.3 Linear Maps -- 2.4 Coordinate Changes -- 2.5 Nonlinear Maps and the Jacobian Matrix -- 2.6 Stable and Unstable Manifolds -- 2.7 Matrix Times Circle Equals Ellipse -- 3 Chaos -- 3.1 Lyapunov Exponents -- 3.2 Chaotic Orbits -- 3.3 Conjugacy and the Logistic Map -- 3.4 Transition Graphs and Fixed Points -- 3.5 Basins of Attraction -- 4 Fractals -- 4.1 Cantor Sets -- 4.2 Probabilistic Constructions of Fractals -- 4.3 Fractals from Deterministic Systems -- 4.4 Fractal Basin Boundaries -- 4.5 Fractal Dimension -- 4.6 Computing the Box-Counting Dimension -- 4.7 Correlation Dimension -- 5 Chaos in Two-Dimensional Maps -- 5.1 Lyapunov Exponents -- 5.2 Numerical Calculation of Lyapunov Exponents -- 5.3 Lyapunov Dimension -- 5.4 A Two-Dimensional Fixed-Point Theorem -- 5.5 Markov Partitions -- 5.6 The Horseshoe Map -- 6 Chaotic Attractors -- 6.1 Forward Limit Sets -- 6.2 Chaotic Attractors -- 6.3 Chaotic Attractors of Expanding Interval Maps -- 6.4 Measure -- 6.5 Natural Measure -- 6.6 Invariant Measure for One-Dimensional Maps -- 7 Differential Equations -- 7.1 One-Dimensional Linear Differential Equations -- 7.2 One-Dimensional Nonlinear Differential Equations -- 7.3 Linear Differential Equations in More than One Dimension -- 7.4 Nonlinear Systems -- 7.5 Motion in a Potential Field -- 7.6 Lyapunov Functions -- 7.7 Lotka-Volterra Models -- 8 Periodic Orbits and Limit Sets -- 8.1 Limit Sets for Planar Differential Equations -- 8.2 Properties of ?-Limit Sets -- 8.3 Proof of the Poincarรฉ-Bendixson Theorem -- 9 Chaos in Differential Equations -- 9.1 The Lorenz Attractor -- 9.2 Stability in the Large, Instability in the Small -- 9.3 The Rรถssler Attractor -- 9.4 Chuaโs Circuit -- 9.5 Forced Oscillators -- 9.6 Lyapunov Exponents in Flows -- 10 Stable Manifolds and Crises -- 10.1 The Stable Manifold Theorem -- 10.2 Homoclinic and Heteroclinic Points -- 10.3 Crises -- 10.4 Proof of the Stable Manifold Theorem -- 10.5 Stable and Unstable Manifolds for Higher Dimensional Maps -- 11 Bifurcations -- 11.1 Saddle-Node and Period-Doubling Bifurcations -- 11.2 Bifurcation Diagrams -- 11.3 Continuability -- 11.4 Bifurcations of One-Dimensional Maps -- 11.5 Bifurcations in Plane Maps: Area-Contracting Case -- 11.6 Bifurcations in Plane Maps: Area-Preserving Case -- 11.7 Bifurcations in Differential Equations -- 11.8 Hopf Bifurcations -- 12 Cascades -- 12.1 Cascades and 4.669201609 -- 12.2 Schematic Bifurcation Diagrams -- 12.3 Generic Bifurcations -- 12.4 The Cascade Theorem -- 13 State Reconstruction from Data -- 13.1 Delay Plots from Time Series -- 13.2 Delay Coordinates -- 13.3 Embedology -- A Matrix Algebra -- A.1 Eigenvalues and Eigenvectors -- A.2 Coordinate Changes -- A.3 Matrix Times Circle Equals Ellipse -- B Computer Solution of Odes -- B.1 ODE Solvers -- B.2 Error in Numerical Integration -- B.3 Adaptive Step-Size Methods -- Answers and Hints to Selected Exercises