Henon Multifractal Map movie.gif [Info on Image]

A Trajectory Through Phase Space in a Lorenz Attractor [Info on Image]

Around and around and around she goes and where she'll land next nobody knows. 

One of the important observations made by Scales and Riper is that developments in warfare do not proceed in a linear way. They mean that, e.g., doubling the number of troops in an engagement does not necessarily double the effectiveness of the military movement because there are always other factors involved. What actually happens depends on other processes, and how they behave all may be sensitive to slight changes in initial conditions. The same kind of greatly varying changes in results depending on slight changes in initial conditions is demonstrated by a class of equations discovered in the course of research on mathematical weather prediction.  (See an exposition of one such equation, the Hénon attractor, here.) When such an equation is calculated through many iterations (a value calculated in one iteration becomes an input value in the next calculation) the answers do not plot close together. If the plotting of the graph immediately above is slowed down one can see that if a researcher were calculating values by hand, it would be a long time before any perceptible pattern would emerge. The sequentially calculated values are far separated and seem to be widely scattered. However, when many hundreds of values are calculated one can see that first a roughly circular pattern emerges and then it switches over to form the rest of a sort of figure-8 pattern, and over time it will go in circles in one half or the other half of the figure.