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AuthorAlligood, Kathleen T. author
TitleChaos [electronic resource] : An Introduction to Dynamical Systems / by Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1997
Connect tohttp://dx.doi.org/10.1007/978-3-642-59281-2
Descript XVII, 603 p. online resource

SUMMARY

BACKGROUND Sir Isaac Newton hrought to the world the idea of modeling the motion of physical systems with equations. It was necessary to invent calculus along the way, since fundamental equations of motion involve velocities and accelerations, of position. His greatest single success was his discovery that which are derivatives the motion of the planets and moons of the solar system resulted from a single fundamental source: the gravitational attraction of the hodies. He demonstrated that the ohserved motion of the planets could he explained hy assuming that there is a gravitational attraction he tween any two ohjects, a force that is proportional to the product of masses and inversely proportional to the square of the distance between them. The circular, elliptical, and parabolic orhits of astronomy were v INTRODUCTION no longer fundamental determinants of motion, but were approximations of laws specified with differential equations. His methods are now used in modeling motion and change in all areas of science. Subsequent generations of scientists extended the method of using differยญ ential equations to describe how physical systems evolve. But the method had a limitation. While the differential equations were sufficient to determine the behavior-in the sense that solutions of the equations did exist-it was frequently difficult to figure out what that behavior would be. It was often impossible to write down solutions in relatively simple algebraic expressions using a finite number of terms. Series solutions involving infinite sums often would not converge beyond some finite time


CONTENT

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 Chuaโ{128}{153}s 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


Mathematics Mathematical analysis Analysis (Mathematics) Statistical physics Dynamical systems Mathematics Analysis Statistical Physics Dynamical Systems and Complexity



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