From Strange Attractor to Period Doubling

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.