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.
- 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.
This is a preview of subscription content, access via your institution.
Tax calculation will be finalised at checkout
Purchases are for personal use onlyLearn about institutional subscriptions
Unable to display preview. Download preview PDF.
© 1982 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
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
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