unanticipated results of analog and digital computer experiments were responsible for most if not all of the discoveries underlying the current era’s revolution in applied nonlinear mathematics Modern Applied Dynamical Systems Emerged from Accidental Computational Discoveries A medical student named Herr, in his thesis research with the “radio engineer’, Van der Pol (1926), was simulating cardiac electrophysiology with an analog device which permitted real time, exploration of a full range of parameter values long before there were fast enough digital processors to do so. Studying the behavior of equations of a periodically, pace maker, driven, nonlinear triode oscillator, Herr found orbital points that appeared to belong to two different periods simultaneously thus violating the uniqueness of solutions of differential equation theory. The Van der Pol relaxation oscillator equations, with their slow buildup and sudden discharge of membrane potential are good models for the slow-fast processes of repolarization and depolarization of Hodgkin-Huxley type equations (Rinzel, 1985). Periodically driven, nonlinear differential equations of the Van der Pol type are generally applicable to the multiplicity of dynamical regimes of neuronal dynamics (Carpenter, 1979; Aihara et al, 1984; Chay and Rinzel, 1985) and, with periodic and aperiodic driving and noise, can be made relevant to particular mammalian neuronal subsystems in the context of clinically relevant global electrophysiological phenomena such as Magoun’s (1954) brain stem evoked EEG and behavioral arousal (Nicolis, 1986; Selz and Mandell, 1992; Mandell and Selz, 1993). In the early 1940’s, using the pre-publication results of similar analog computer studies (Levinson, 1949), the Cambridge mathematicians, Mary Cartwright and Joe Littlewood (1945; McMurran, S., Tattersal, J.,1999) used geometric methods to prove that the highly nonlinear, periodically driven Van der Pol equations, depending upon one or two changing