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random” patterns. We now know such phenomena to be universal characteristics of bifurcation scenarios in nonlinear dynamical systems where bifurcation means discontinuous changes in patterns of behavior (dependent variables) resulting from smooth changes in parameters (independent variables). Alerted to their presence in computer experiments with biologically relevant nonlinear differential equations, these phenomena have since been found in time series from patch clamped membrane channels, single neurons, neuronal networks, neuroendocrine systems, brain waves and patterns of behavior in animals and man (see below). Cartwright-Littlewood found that the inner and outer edges of the domains of attraction (all the initial values that eventually wind up in the attractor—the limit set of all bounded solutions) of two different sets of subharmonic periods for the same parameter settings were interlaced at many scales in what is today called a fractal basin boundary. It was in this way that the specific values of the end state are understood to be indeterminate since the starting values in the fractal basin boundary are impossible to isolate and specify with adequate experimental precision.

Similar biologically-relevant analog computer discoveries about the Van der Pol and comparable periodically forced, dissipative (energy utilizing) Duffing equations (Zeeman, 1976) were made in the early 1960’s by electrical engineer, Yoshi Ueda (1992), but his thesis director, Chihiro Hayashi of Japan’s Kyoto University, was sufficiently disturbed by this evidence for the existence of bounded solutions (attractors) that were neither fixed points (equilibria) nor periodic orbits (cycles), the only ones known at the time and therefore “strange,” that he refused to let Ueda publish his findings until he did so as an independent investigator in the 1970’s.

In the early 1960’s, Edward Lorenz (1963), a meteorologist and student of the Harvard mathematician and dynamical systems pioneer, George Birkoff (1922), was computing the output of a very reduced subset of Saltzman’s differential equations for predicting the weather (1962). Lorenz found that numerically integrated trajectories manifested unpredictable times and directions of motion between the two spiral orbits of what has come to be known as the Lorenz attractor. Very small differences in starting values led to widely diverse final values, and, just

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