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  • © 1997

Chaos

An Introduction to Dynamical Systems

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Part of the book series: Textbooks in Mathematical Sciences (TIMS)

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  • ISBN: 978-3-642-59281-2
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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xvii
  2. One-Dimensional Maps

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 1-42
  3. Two-Dimensional Maps

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 43-104
  4. Chaos

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 105-147
  5. Fractals

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 149-191
  6. Chaos in Two-Dimensional Maps

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 193-230
  7. Chaotic Attractors

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 231-271
  8. Differential Equations

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 273-327
  9. Periodic Orbits and Limit Sets

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 329-358
  10. Chaos in Differential Equations

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 359-397
  11. Stable Manifolds and Crises

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 399-445
  12. Bifurcations

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 447-498
  13. Cascades

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 499-536
  14. State Reconstruction from Data

    • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
    Pages 537-556
  15. Back Matter

    Pages 557-603

About this book

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 differ­ 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.

Keywords

  • Eigenvalue
  • approximation
  • behavior
  • bifurcation
  • calculus
  • derivative
  • differential equation
  • eigenvector
  • fixed-point theorem
  • hamiltonian system
  • integration
  • manifold
  • stability
  • system
  • time

Authors and Affiliations

  • George Mason University, USA

    Kathleen T. Alligood, Tim D. Sauer

  • University of Maryland, USA

    James A. Yorke

Bibliographic Information

Buying options

eBook
USD 74.99
Price excludes VAT (USA)
  • ISBN: 978-3-642-59281-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout