Discrete Dynamical Modeling and Analysis of the R-S Flip-Flop Circuit

TL;DR

This paper introduces a discrete dynamical model for the R-S flip-flop circuit that captures its complex behaviors, including chaos and periodicity, validated through comparison with differential equation models.

Contribution

It presents a novel discrete model that replicates the qualitative dynamics of the R-S flip-flop, bridging discrete and continuous circuit models.

Findings

Model exhibits chaos and high-period orbits

Dynamics depend on system parameters

Validation aligns with differential equation models

Abstract

A simple discrete planar dynamical model for the ideal (logical) R-S flip-flop circuit is developed with an eye toward mimicking the dynamical behavior observed for actual physical realizations of this circuit. It is shown that the model exhibits most of the qualitative features ascribed to the R-S flip-flop circuit, such as an intrinsic instability associated with unit set and reset inputs, manifested in a chaotic sequence of output states that tend to oscillate among all possible output states, and the existence of periodic orbits of arbitrarily high period that depend on the various intrinsic system parameters. The investigation involves a combination of analytical methods from the modern theory of discrete dynamical systems, and numerical simulations that illustrate the dazzling array of dynamics that can be generated by the model. Validation of the discrete model is accomplished by comparison with certain Poincar\'e map like representations of the dynamics corresponding to three-dimensional differential equation models of electrical circuits that produce R-S flip-flop behavior.