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Dynamical Systems

Examples of Complex Behaviour

  • Textbook
  • © 2005

Overview

Authors:
  1. Jürgen Jost

    1. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

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  • Breadth of scope is unique
  • Unlike many recent textbooks on chaotic systems no superficial treatment but explanations of the deep underlying mathematical ideas
  • No technical proofs but an introduction to the whole field that is based on the specific analysis of carefully selected examples
  • Includes a section on cellular automata
  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

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Table of contents (8 chapters)

  1. Front Matter

    Pages I-VIII
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  2. Introduction

    Pages 1-6

  3. Stability of dynamical systems, bifurcations, and generic properties

    Pages 7-60

  4. Discrete invariants of dynamical systems

    Pages 61-98

  5. Entropy and topological aspects of dynamical systems

    Pages 99-110

  6. Entropy and metric aspects of dynamical systems

    Pages 111-117

  7. Entropy and measure theoretic aspects of dynamical systems

    Pages 119-152

  8. Smooth dynamical systems

    Pages 153-168

  9. Cellular automata and Boolean networks as examples of discrete dynamical systems

    Pages 169-180

  10. Back Matter

    Pages 181-189
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Keywords

About this book

Our aim is to introduce, explain, and discuss the fundamental problems, ideas, concepts, results, and methods of the theory of dynamical systems and to show how they can be used in speci?c examples. We do not intend to give a comprehensive overview of the present state of research in the theory of dynamical systems, nor a detailed historical account of its development. We try to explain the important results, often neglecting technical re?nements 1 and, usually, we do not provide proofs. One of the basic questions in studying dynamical systems, i.e. systems that evolve in time, is the construction of invariants that allow us to classify qualitative types of dynamical evolution, to distinguish between qualitatively di?erent dynamics, and to studytransitions between di?erent types. Itis also important to ?nd out when a certain dynamic behavior is stable under small perturbations, as well as to understand the various scenarios of instability. Finally, an essential aspect of a dynamic evolution is the transformation of some given initial state into some ?nal or asymptotic state as time proceeds. Thetemporalevolutionofadynamicalsystemmaybecontinuousordiscrete, butitturnsoutthatmanyoftheconceptstobeintroducedareusefulineither case.

Reviews

"This book provides a survey of various topics of dynamical systems. Applications of both the concepts and the results are presented.

The author takes the opportunity to explain the underlying fundamental mathematical concepts involved in the results, for example the Conley-Floer theory, which is a topic that is not commonly studied in introductory texts on dynamical systems. The book studies entropy and related concepts within topological, metric, measure theoretic and smooth settings, giving connections with information theory, cellular automata and Boolean networks.

The book is written in a careful style. Most of the results are given without proof, though the necessary references for them are included. Many of the results are illustrated through carefully chosen examples."  (Sergio Plaza, Mathematical Reviews) 

Authors and Affiliations

  • Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Jürgen Jost

About the author

Jürgen Jost is a widely-known and successful textbook author. His textbooks with Springer see related literature.

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