as importantly, far apart initial values could be found close together in the limit set

as importantly, far apart initial values could be found close together in the limit set. This behavior was called “sensitivity to initial conditions” by David Ruelle (1978; Ruelle and Takens, 19771). It is noteworthy, however, that over a range of values of the parameters, the overall pattern of the orbits of the Lorenz attractor results in characteristic geometric pictures as well as invariant statistical descriptors. Qualitative and quantitative global similarities were gained while specific solutions were lost in these “strange attractors” of nonlinear systems. Analog computer simulation of a simpler set of equations inspired by nonlinear chemical reaction kinetics led to the discovery by Rdéssler (1976) of another early and generic strange attractor combining sensitivity to initial conditions and characteristic geometries and measures. It was the Russian mathematicians, A.N. Kolmogorov (1957), Sinai (1959) and V.I. Arnold (Arnold and Avez, 1968), the French mathematicians, Rene Thom (1972) and David Ruelle (1978) and the U.C. Berkeley mathematicians, Steve Smale (1967) and his student, Rufus Bowen (1975), and their associates who gathered together these and other related computational discoveries and embedded them in a qualitative theory of nonlinear differential equations, using a variety of formalisms, including point set and differential topology, geometry, analysis and ergodic (having an invariant statistical description) measure theory that formally established the fundamentals for research in nonlinear dynamical systems. Here a dynamical system refers generally and simply to the components and nonlinear processes (transformations) that move points (values) in discrete (“map”) or continuous (“differential equation”) time around in an appropriately defined space. The phrase, “nonlinear transformation” in this context does not imply easily solvable curved functions, such as those representing the sigmoid kinetic or threshold functions of enzymes and neuronal networks or those that smoothly log transform the amplitudes of auditory or other sensory modalities in man, but rather allude to expressions containing products, powers and functions of the computational and/or experimental variables x,, such as x,x,, (x,)° Orsin(x) . 191 HOUSE_OVERSIGHT_013691