Jules Henri Poincare was dubbed by E. T. Bell as the "Last Universalist", a man who is at ease in all branches of mathematics, both pure and applied. Poincare was one of these rare savants who was able to make many major contributions to such diverse fields as analysis, algebra, topology, astronomy, and theoretical physics. Like Gauss, another universalist, his mind was constantly brimming with highly creative ideas, but unlike Gauss, he published extensively. Among his many published works are several highly readable popular pieces that tried to give the general public a flavor of the workings of science.
The ideas of dynamical chaos was first glimpsed by Poincare when he entered a contest sponsored by the king of Sweden. One of the questions in this contest was to show rigorously that the solar system as modeled by Newton's equations is dynamically stable. The question was nothing more than a generalization of the famous three body problem, which was considered one of the most difficult problems in mathematical physics. In essence, the three body problem consists of nine simultaneous differential equations (all linear, each of second order). The difficulty was in showing that the general solution converges since any solution will be given in terms of infinite series. While Poincare did not succeed in giving a complete solution, he made such a major headway in attacking the problem that he was awarded the prize anyway. The distinguished Weierstrass, who was one of the judges, said, "this work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics." A lively account of this event is given in Newton's Clock: Chaos in the Solar System.
To show how visionary Poincare was, it is perhaps best if he described the Hallmark of Chaos - sensitive dependence on initial conditions - in his own words:
If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation appproximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. - in a 1903 essay "Science and Method"
A biography that gives more information about his other mathematical achievements besides chaos can be found in this nice History of Mathematics site under Poincare.