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From Strange Attractor to Period Doubling

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Part of the Applied Mathematical Sciences book series (AMS,volume 41)

Abstract

In Chapter 3 we argued that there was a whole range of r-values (near r = 28.0) for which the Lorenz equations possessed a strange attractor. We did not expect to find stable periodic orbits for any r-values in this range. In Chapter 4 we studied a very different range of r-values. In this range we did find stable periodic orbits. The purpose of this chapter is to show how the behaviour changes from strange attractor to period doubling windows as r increases. We will first examine the problem by studying return maps. Then we will ask how well we can model the Chapter 4 type behaviour with a one-dimensional discrete map of an interval to itself, and discuss the difficulties of this approach. Finally we shall work towards a global understanding of the Lorenz equations which will be useful when we want to know how the Lorenz equations behave for parameter values other than σ = 10 and b = 8/3, and which shows how strange attractor and period doubling fit together in a more general context.

Keywords

  • Periodic Orbit
  • Unstable Manifold
  • Homoclinic Orbit
  • Stable Manifold
  • Strange Attractor

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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  • DOI: 10.1007/978-1-4612-5767-7_5
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© 1982 Springer-Verlag New York Inc.

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Sparrow, C. (1982). From Strange Attractor to Period Doubling. In: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Applied Mathematical Sciences, vol 41. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5767-7_5

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  • DOI: https://doi.org/10.1007/978-1-4612-5767-7_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-90775-8

  • Online ISBN: 978-1-4612-5767-7

  • eBook Packages: Springer Book Archive