
For almost all of its history, applied mathematics has been concerned with the solution of linear differential equations; most physical phenomena seemed to be approximately governed by such equations (see linearity). In such systems, the effect of a small change to the initial conditions makes only a small change to the system as a whole (for example, moving a snooker ball one millimetre makes little difference to the position in which it comes to rest after being struck by the cueballâ€”all other things being equal).
However, during the 1970s and 1980s it gradually became apparent that for many systems (in particular, the Earth\'s weather), linear differential equations would not adequately model the system, particularly for some initial conditions. So attention was shifted to the use of nonlinear differential equations, which predicted that small changes in the original conditions could sometimes make vast differences to the system. The famous example of this is that of a butterfly fluttering its wings and thereby causing a hurricane on the other side of the globe. (This does not, of course, happen every time a butterfly flutters its wings; it must be in exactly the right place at exactly the right time.)
The difficulty in this approach is that practically no nonlinear equations can be solved with the methods of analysis. This has meant that the only way solutions can be found is through the use of computers that work out the effect of the equations over very short periods of time, and find approximate solutions. This has to be done very quickly, which means that computer scientists working in chaos theory are often pressing right up against the limits of the machines and software available today.
At the time of writing (early 1990s), chaos theory remains somewhat esoteric, of more significance in computing, mathematics and the physical sciences than in the world at large. There are signs, however, that it is beginning to make a mark in the arts, as part of the new pluralism of postmodernism. In the 1980s, artistic creators of all kinds began to explore the idea that chaos theory provided a rationale by which artists, instead of trying to impose order and structure, could exploit the random and unexpected. Examples range from the work of Boulez and his assistants at IRCAM (blending randomness with strict musical serialism) to improvised comedy in the theatre (where there was a huge 1990s upsurge throughout the Western world), from â€˜computer poetryâ€™ (in which the â€˜poetâ€™ programs the computer to create random wordpatterns, and then selects from the results) to the work of such architects as Isozaki and Kuramata in Japan and Frank Gehry in California.
Work on the chaos theory and its implications is advancing so fast that it is impossible to predict future developments indeed to predict anything except that predictions and descriptions of the current state of the theory will be out of date almost as soon as made. KMcL SMcL 
