I'm going to wind up with chaos and climate, but the route starts with numerical weather prediction. Numerical weather prediction itself starts much farther back than most people realize -- now about 90 years old. And it didn't start with the simplest possible weather prediction model. If you're not up on your history of science and technology, you didn't bat an eyelash at my mention of numerical weather prediction being 90 years old. Electronic computers are only 60-70. 90+ years ago, when the first numerical weather prediction (NWP) was done, 'computer' meant a person who, with pen and paper, slogged through the calculations.
The first NWP was performed by hand, by Lewis Frye Richardson. He did so in between ambulance runs during World War I. The reason he was in an ambulance, rather than the much safer trenches, is that he was a conscientious objector to warfare (Society of Friends (Quaker) by religion). Nevertheless, he did survive the war and completed his numerical prediction. It was finally published in 1922, with it having been essentially completed in 1919.
The model he used was what we now call a primitive equation model. It took the laws of conservation of mass, energy, and momentum in their full complexity and tried to solve them. The first successful numerical weather prediction was not made until about 1948, published in 1950: Charney, J. G, R. Fjortoft, and J. von Neumann, "Numerical Integration of the Barotropic Vorticity Equation", Tellus, 4, 237-254, 1950. It was done on a much simpler equation -- bearing much the same resemblance to the primitive equations as the simplest climate model I've mentioned before does to a full complexity climate model. This model (which Richardson could have done by hand more easily than what he took on) was run on one of the first electronic computers -- ENIAC.
The first model to implement something comparable to what Richardson tackled was not done for another 2 decades after the ENIAC model (the '6 layer PE' model Shuman, F. G., and J. B. Hovermale, An Operational Six Layer Primitive Equation Model", J. Applied Meteorology, 7, 525-547, 1968).
In any case, Richardson's forecast has often been called a glorious failure in the ensuing decades. The failure part being that the forecast was so drastically in error -- predicting a surface pressure change of over 100 mb, when only 6 or so would have been considered large. He did recognize the likely sources of his problem, but doing this kind of computation by hand was too expensive (in time) to do multiple trials to nail down exactly which was the source of his problem and which idea for repair would take care of his problem.
The glorious aspect is his enduring contribution to the field. For starters, he invented numerical weather prediction. Much of what is done today is still dependent on approaches he invented. He also foresaw massively parallel processing, and one of the central problems in that -- making sure that your different processing 'nodes' (people, in his case, computer processors in ours) remained synchronized.
For more on Richardson's forecast, see especially Weather Prediction by Numerical Process, by Lewis F. Richardson. Original publication in 1922, republished in 1965 by Dover Publications. This is the original, full, document. More recent consideration was made by G. W. Platzman, "Richardson's Weather Prediction", Bull. Amer. Meteor. Soc., 60, 302-312, 1968. Platzman includes discussion of the sources of Richardson's problems. There is a later note in BAMS by Platzman giving more considerations and ideas.