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1982 | Buch

Introduction and Simple Properties Kapitel lesen Erstes Kapitel lesen

The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors

verfasst von: Colin Sparrow

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Professional Book Archive

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Über dieses Buch

The equations which we are going to study in these notes were first presented in 1963 by E. N. Lorenz. They define a three-dimensional system of ordinary differential equations that depends on three real positive parameters. As we vary the parameters, we change the behaviour of the flow determined by the equations. For some parameter values, numerically computed solutions of the equations oscillate, apparently forever, in the pseudo-random way we now call "chaotic"; this is the main reason for the immense amount of interest generated by the equations in the eighteen years since Lorenz first presented them. In addition, there are some parameter values for which we see "preturbulence", a phenomenon in which trajectories oscillate chaotically for long periods of time before finally settling down to stable stationary or stable periodic behaviour, others in which we see "intermittent chaos", where trajectories alternate be­ tween chaotic and apparently stable periodic behaviours, and yet others in which we see "noisy periodicity", where trajectories appear chaotic though they stay very close to a non-stable periodic orbit. Though the Lorenz equations were not much studied in the years be­ tween 1963 and 1975, the number of man, woman, and computer hours spent on them in recent years - since they came to the general attention of mathematicians and other researchers - must be truly immense.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction and Simple Properties
Abstract
In 1963 E. N. Lorenz wrote a remarkable paper. In it he described a three parameter family of three-dimensional ordinary differential equations which appeared, when integrated numerically on a computer, to have extremely complicated solutions. These equations, now known as the Lorenz equations, have been studied by many authors in the years since 1963 and one of our aims, in these notes, is to contribute to this study. It is necessary, therefore, to explain some of the reasons why the equations generated so much interest initially and why they warrant further study.
Colin Sparrow
Chapter 2. Homoclinic Explosions: The First Homoclinic Explosion
Abstract
When r > 1 there is a two-dimensional sheet of initial values in R3 from which trajectories tend towards the origin. This two-dimensional sheet is called the stable manifold of the origin. Near the origin we know that this sheet looks like a plane (the plane associated with the two negative eigenvalues of the flow linearized near the origin) and if we wished we could approximate its position elsewhere by integrating the equations in backwards time with initial conditions lying on this plane and near to the origin. It appears that when r is only moderately larger than one, the stable manifold of the origin divides R3 into two halves in a fairly simple way. Trajectories started in one half-space tend towards C1 and trajectories started in the other half-space tend towards C2. Trajectories started on the stable manifold of the origin tend, of course, towards the origin. (See Fig. 2.1.)
Colin Sparrow
Chapter 3. Preturbulence, Strange Attractors and Geometric Models
Abstract
We commence our study of the parameter region r > 13.926 by attempting to confirm the first of the two suggestions made at the end of the last chapter. We can use the computer to locate stable and non-stable periodic orbits. The techniques used are based on Newton1s method, are well known, and are described in Appendix E. To use the techniques it is necessary to have a reasonably good guess for the position and period of the orbit we wish to locate. Appendix E contains practical hints on how to come by such initial guesses, and on how to use the techniques. Appendix E also contains the description of a technique for following a periodic orbit with changing parameter.
Colin Sparrow
Chapter 4. Period Doubling and Stable Orbits
Abstract
Several authors have noticed that numerical simulations of the Lorenz equations indicate the existence of stable periodic orbits in some intervals of r-values. This behaviour is quite different from the behaviour discussed in Chapter 3, since for r-values near 28.0 we saw no stable periodic orbits, and had strong arguments that none could exist. We will attempt to reconcile the two phenomena in Chapter 5.
Colin Sparrow
Chapter 5. 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.
Colin Sparrow
Chapter 6. Symbolic Description of Orbits: The Stable Manifolds of C1 and C2
Abstract
At various points in the last four chapters we have found it useful to be able to describe periodic orbits and trajectories with sequences of two symbols. When we studied homoclinic explosions in Chapter 2, we saw that we were rigorously justified in describing the periodic orbits and trajectories born (or destroyed) in these bifurcations with sequences of two symbols; each symbol corresponded to one of the two tubes surrounding (at the critical r-value) the two branches of the homoclinic orbit, and trajectories were assigned symbolic sequences according to the order in which they journeyed through these tubes. These descriptions were only local. In Chapters 3 through 5, we found we had need for a global method of describing orbits and trajectories. In Chapters 3 and 5, we defined symbol sequences according to the order in which trajectories intersected two halves of some suitable return surface. In Chapter 4, we assigned symbol sequences to periodic orbits according to whether their successive local maxima in the z-variable lay in x > 0 or x < 0. These two methods appeared to be equivalent.
Colin Sparrow
Chapter 7. Large r
Abstract
Numerical experiments indicate that for σ = 10, b = 8/3 and r > 313, there is a stable symmetric xy periodic orbit. Furthermore, we suggested in Chapter 5 that this periodic orbit and the three stationary points would, for large enough r, make up the whole of the non-wandering set. In this chapter, we show that there are theoretical reasons to expect both of these results. At the same time, we show that qualitatively more complicated large r behaviour may be expected for some values of the parameters σ and b.
Colin Sparrow
Chapter 8. Small b
Abstract
Our investigation of the behaviour of the Lorenz equations for large r (Chapter 7) suggested that we would see qualitatively more complicated behaviour when the parameter b was small. We now investigate some aspects of this behaviour.
Colin Sparrow
Chapter 9. Other Approaches, Other Systems, Summary and Afterword
Abstract
In this section we will not summarize all the results discussed in these notes (readers are referred to the relevant chapters for such summaries) but will attempt to draw together all the material that is useful in predicting bifurcations for general values of the parameters σ, b and r. We shall concentrate first on the two-dimensional parameter plane given by σ = 10 on which we have the most experience.
Colin Sparrow
Backmatter
Metadaten
Titel
The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors
verfasst von
Colin Sparrow
Copyright-Jahr
1982
Verlag
Springer New York
Electronic ISBN
978-1-4612-5767-7
Print ISBN
978-0-387-90775-8
DOI
https://doi.org/10.1007/978-1-4612-5767-7