Dynamical Systems

Dynamical Systems : Differential Equations, Maps and Chaotic Behaviour

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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 behaviour. The unprecedented popular interest shown in recent years in the chaotic behaviour of discrete dynamic systems including such topics as chaos and fractals has had its impact on the undergraduate and graduate curriculum. The book is aimed at courses in dynamics, dynamical systems and differential equations and dynamical systems for advanced undergraduates and graduate students. Applications in physics, engineering and biology are considered and introduction to fractal imaging and cellular automata are given.
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Product details

  • Hardback | 330 pages
  • 165.1 x 241.3 x 19.05mm | 589.67g
  • London, United Kingdom
  • English
  • diagrams, bibliography, index
  • 0412390701
  • 9780412390708

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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