Threshold voltage dynamics of chaotic RS flip-Flops

Highlights: • Mathematical exploration of possible threshold voltage models. • Piecewise-linear perturbations leading to chaos. • Continuous extensions of Boolean models. - Abstract: Chaotic Set/Reset (RS) flip-flop circuits are investigated once again in the context of discrete planar dynamical system models of the threshold voltages, but this time starting with simple bilinear (minimal) component models derived from first principles. The dynamics of the minimal model is described in detail, and shown to exhibit some of the expected properties, but not the chaotic regimes typically found in simulations of physical realizations of chaotic flip-flop circuits. Any electronic physical realization of a chaotic logical circuit must necessarily involve small perturbations from the ideal - usually with large or even nonexistent derivatives in small diameter subsets of the phase space. Therefore, perturbed forms of the minimal model are also analyzed in considerable detail. It is proved that very slightly perturbed minimal models can exhibit chaotic regimes, sometimes associated with chaotic strange attractors, as well as some of the bifurcations present in most of the differential equations models for similar physical circuit realizations. In essence, this work is a mathematical exploration of simple models that reproduce the qualitative behavior of threshold control units of a chaotic RS flip-flop design. It is also shown that this method can be extended to other similar circuits. Validation of the approach developed is provided by some comparisons with (mainly simulated) dynamical results obtained from more traditional investigations.