Dynamical Systems

Dynamical Systems : Differential Equations, Maps, and Chaotic Behaviour

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Description

This text discusses the qualitative properties of dynamical systems including both differential equations and maps. The approach taken relies heavily on examples (supported by extensive exercises, hints to solutions and diagrams) to develop the material, including a treatment of chaotic behavior.

The unprecedented popular interest shown in recent years in the chaotic behavior of discrete dynamic systems including such topics as chaos and fractals has had its impact on the undergraduate and graduate curriculum. However there has, until now, been no text which sets out this developing area of mathematics within the context of standard teaching of ordinary differential equations.

Applications in physics, engineering, and geology are considered and introductions to fractal imaging and cellular automata are given.
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Product details

  • Paperback | 330 pages
  • 156 x 235 x 19.3mm | 499g
  • CRC Press
  • London, United Kingdom
  • English
  • diagrams, bibliography, index
  • 0412390809
  • 9780412390807

Table of contents

Part 1 Introduction: preliminary ideas; autonomous equations; autonomous systems in the plane; construction of phase portraits in the plane; flows and evolution. Part 2 Linear systems: linear changes of variable; similarity types for 2x2 real matrices; phase portraits for canonical systems in the plane; classification of simple linear phase portraits in the plane; the evolution operator; affine systems; linear systems of dimension greater than two. Part 3 Non-linear systems in the plane: local and global behaviour; linearization at a fixed point; the linearization theorem; non-simple fixed points; stability of fixed points; ordinary points and global behaviour; first integrals; limit points and limit cycles; Poincare-Bendixson theory. Part 4 Flows on non-planar phase spaces: fixed points; closed orbits; attracting sets and attractors; further integrals. Part 5 Applications I - planar phase spaces: linear models; affine models; non-linear models; relaxation oscillations; piecewise modelling. Part 6 Applications II - non-planar phase spaces, families of systems and bifurcations: the Zeeman models of heart beat and nerve impulse; a model of animal conflict; families of differential equations and bifurcations; a mathematical model of tumor growth; some bifurcations in families of one-dimensional maps; some bifurcations in families of two-dimensional maps; area-preserving maps, homoclinic tangles and strange attractors; symbolic dynamics; new directions.
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