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Phragmen's queries and Poincare's response are described in Andersson (1994). Poincare Les methodes nouvelles, Section 397. More on Poincare's discovery of homoclinic tangles can be found in Holmes (1990) and Diacu and Holmes (1996), the latter being less demanding mathematically.

mechanics), the qualitative behaviour of the solutions often was highly complex, with extreme sensitivity to initial data. The deterministic nature of the equations did not imply that accurate predictions were possible. In his 1903 essay Science and Method, he incorporated this fundamental property of dynamical systems into his definition of chance:

A very slight cause, which escapes us, determines a considerable effect which we cannot help seeing, and then we say this effect is due to chance. If we could know exactly the laws of nature and the situation of the universe at the initial instant, we should be able to predict exactly the situation of this same universe at a succeeding instant. But even when the natural laws should have no further secret for us, we could know the initial situation only approximately. If that permits us to foresee the subsequent situation with the same degree of approximation, this is all we require, we say the phenomenon has been predicted, that it is ruled by laws. But this is not always the case; it may happen that slight differences in the initial conditions produce very great differences in the final phenomena; a slight error in the former would make an enormous error in the latter. Prediction becomes impossible and we

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