concerning the onset of turbulence, as computer simulations of three or more coupled t-o < -s nonlinear first-order ordinary differential equations have led to chaotic solutions. ' .. I" P^ ^i P Such an observation led Ruelle and Takens (1971) to hypothesize that transition to V 2 < c w c turbulence can be quantitatively explained by deterministic equations. Their work co H < fl l/j £ has produced exciting and profound results and the resulting theory has become 5, § rwidely known as the

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... Takens theory of turbulence. They showed mathematically that, for a nonlinear initial-boundary value problem, with a large critical parameter such as Re, chaos resulted from repeated bifurcations of the solution.