The peculiar result of the Lorenz equations is that they produce deterministic chaos. The problem is deterministic, because we know everything there is about how it will instantaneously change (there is nothing random about these equations). For high enough Rayleigh numbers, however, it is chaotic because even small changes in the initial conditions can lead to very different behavior at long times because the small differences grow in a non-linear feedback with time. This is known as the Butterfly effect because Lorenz suggested that a butterfly flapping in one part of the world might make it impossible to predict the weather a week in advance.
Nevertheless, Chaos does not mean random unpredictability. After watching these equations flip and flop for a while, you can see that in some respects they are fairly well behaved. The overall behavior is restricted to lie in the Strange attractor (the system can't suddenly stop or start spinning a thousand times faster) and the overall patterns repeat in a quasi-periodic fashion. The problem is that you can't follow the problem for an arbitrarily long period of time.
But watching these equations perform, you might get the feeling that you should be able to follow them for some period of time. If you consider each wobble in the problem a cycle, then Lorenz actually showed that you can actually predict the behavior of the equations at least one cycle ahead. Here we will reproduce that result (and more) and introduce some of the basics of non-linear forecasting.
Here is the basic problem. Suppose that you were handed a time-series of (or anything else, maybe rainfall in Arizona). You might want to ask yourself ``how much information is in this time-series...if I know a part of it, how much into the future can I predict?'' This is the basic question of non-linear forecasting (e.g. [8, 9]). If you're data is produced by a low dimensional chaotic attractor, the answer may be that you can predict more than you thought.
What Lorenz asked was ``if I know the peak value of at some time, can I predict the value of the next peak?'' To do this he first formed a series of all the maximum values of with time (i.e. he had a list of peak1,peak2,...,peakN), then for each peak i he made a plot of peak i+1 as a function of peak i (easier to show than say). And he found the following remarkable picture known as The Lorenz Map
Figure 8: The Lorenz Map for predicting the height of the next peak in given knowledge of the current height. Except for the narrow peak near , the narrowness of this graph suggests that you can predict the height of the next peak extremely well.
To answer that, do the following problems