The mathematical theory of chaos

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Abstract

The basic concepts of the mathematical theory of chaos are presented through a brief analysis of some interesting dynamical systems in one-, two- and three-dimensional space. We start with a discussion of interval maps and observe that when such maps are monotonic, their iterates behave in an orderly fashion. Then, by way of contrast, we study a well-known quadratic1 map iterates clearly manifest the archetypal characteristics of chaos, such as period-doubling bifurcations and the existence of a strange attractor. As a means of indicating that mappings in two dimensions yield a richer variety of chaotic regimes than do interval maps, we next discuss the horseshoe and solenoidal mappings of the two-disk. Dizzying forms of chaos emerge from these mappings, but there is an irony-the chaotic behavior can be characterized in an orderly way. We conclude with a cursory examination of the Lorenz differential equation in three-space: a primary source of the recent interest in chaos theory.

Original languageEnglish (US)
Pages (from-to)1039-1045
Number of pages7
JournalComputers and Mathematics with Applications
Volume12
Issue number3-4 PART 2
DOIs
StatePublished - 1986

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

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