The Lorenz Strange Attractor

Table Of Contents

• Strange Attractors
• The Lorenz Strange Attractor
• References

Strange Attractors

First, we need to set the stage with the idea of a dynamical system. Instead of a formal definition, I think a few examples will convey the idea best. Think about the movement of a pendulum, a planet and moon orbiting a star in our solar system, or a bouncing ball on your kitchen floor. An attractor for such a system is the behavior, or set of behaviors, the system will approach after a long enough time. In the case of the bouncing balls, the attractor is quite simple -- it is the flat surface of the floor. Attractors don't need to be so simple. Consider for example the orbiting planet. In this case, the attractor might be complex shape. Now, a strange attractor is an attractor that is "complicated" - a non-integer Hausdorff dimension for example.

The thing that makes strange attractors so fascinating is that they show the overall patterns of a dynamical system in spite of the fact that the systems described are unbelievably complex. The way a body orbits three or more gravitating bodies is a very good example. Another good example is the strange attractor attributed to Lorenz discussed on this page.

The Lorenz Strange Attractor

Lorenz was studying weather prediction, and he developed a rather simple model involving a differential equation in three dimensional Euclidean space. He had great difficulty solving this equation numerically because it is very sensitive to its initial conditions. Each time he would solve the system, even with tiny rounding changes in the 6th digit, he would obtain radically different solutions. This sort of sensitivity has come to be part of the definition of chaotic systems. The differential equation he was working with was:

or

with the following initial condition:

while not part of the system proper, a common set of parameters are:

The picture below is a plot of this equation projected onto the XY-plane. The equation was solved using a short C program implementing a very simple version of Euler's method and was graphed with Povray. While Povray may appear to be a bit of overkill for such a simple graph, the flexibility gained for further investigation of the curve is very invaluable. The Povray input file can be found here. If you don't have a C compiler, shame on you, here is a Perl version. If you don't have Perl, then double shame on you. :)

Most often people try to use Euler's method to solve this equation; however, a more sophisticated solution method is required to really good plots of the curve. An adaptive Euler algorithm is good enough to do the job, and more advanced techniques like RK don't generally yield better results. The picture below was generated with Povray like the one above, but it has a different point of view and no cylinders are used to connect the spheres. The Povray input file can be found here. The Povray include file was generated with a non-standard, adaptive version of Euler's algorithm that attempts to bound the magnitude of the delta on f at each step. Here is a C program implementing this algorithm. Again, if you don't have C, you can find a Perl version here. Without this kind of bounding, the spheres making up the picture would not form a smooth intersection. In fact, adjacent spheres would not even touch in some cases.

Click to enlarge

Click here for a movie (about 1MB)

References

For a very interesting introduction to the history of Chaos I would suggest James Glick's book: "CHAOS: Making a New Science" (ISBN: 0-14-009250-1). This book has almost no mathematics or formal material regarding Fractals or dynamical systems, but is well worth the read for the historical perspective.

If you are looking for a good introduction to dynamical systems, I would suggest Steven Storgatz's book: "Nonlinear Dynamics and Chaos" ISBN 0-7382-0453-6).

For a good, encyclopedic, introduction to the field in general, I strongly suggest "Chaos and Fractals: New Frontiers of Science" by Peitgen, Jurgens, and Saupe. This book is notable because of it's clear presentation and breadth of coverage. It's a great book to have around for the casual reader because it is broken up into semi self-contained sections that one can just pick up and read.