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Applied Computational Physics$
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Joseph F. Boudreau and Eric. S. Swanson

Print publication date: 2017

Print ISBN-13: 9780198708636

Published to Oxford Scholarship Online: February 2018

DOI: 10.1093/oso/9780198708636.001.0001

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Nonlinear dynamics and chaos

Nonlinear dynamics and chaos

(p.424) 13 Nonlinear dynamics and chaos
Applied Computational Physics

Joseph F. Boudreau

Eric S. Swanson

Oxford University Press

Simple maps and dynamical systems are used to explore chaos in nature. The discussion starts with a review of the properties of nonlinear ordinary differential equations, including the useful concepts of phase portraits, fixed points, and limit cycles. These notions are developed further in an examination of iterative maps that reveal chaotic behavior. Next, the damped driven oscillator is used to illustrate the Lyapunov exponent that can be used to quantify chaos. The famous KAM theorem on the conditions under which chaotic behavior occurs in physical systems is also presented. The principle is illustrated with the Hénon-Heiles model of a star in a galactic environment and billiard models that describe the motion of balls in closed two-dimensional regions.

Keywords:   iterative maps, nonlinear differential equations, chaos, Lyapunov exponent, Hénon-Heiles model, KAM theorem

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