Qualitative models and experimental investigation of chaotic NOR gates and set/reset flip-flops

Abstract

It has been observed through experiments and simulations that logical circuits based upon Chua's circuit exhibit complex dynamical behaviour. This behaviour can be used to design analogues of more complex logic families and some properties can be exploited for electronics applications. Some of these circuits have been modelled as systems of ordinary differential equations. However, as the number of components in newer circuits increases so does the complexity. This renders continuous dynamical systems models impractical and necessitates new modelling techniques. In recent years, some discrete dynamical models have been developed using various simplifying assumptions. To create a robust modelling framework for chaotic logical circuits, we developed both deterministic and stochastic discrete dynamical models, which exploit the natural recurrence behaviour, for two chaotic NOR gates and a chaotic set/reset flip-flop. This work presents a complete applied mathematical investigation of logical circuits. Experiments on our own designs of the above circuits are modelled and the models are rigorously analysed and simulated showing surprisingly close qualitative agreement with the experiments. Furthermore, the models are designed to accommodate dynamics of similarly designed circuits. This will allow researchers to develop ever more complex chaotic logical circuits with a simple modelling framework.

Standard map-like models for single and multiple walkers in an annular cavity

“…To prove the map ( 4) is chaotic we use the main theorem of Li and Yorke [16], and to satisfy the hypothesis of the theorem we search for 3-cycles. While a cardinality argument may work [17], it would be quite tricky to keep track of the intersections between f 3 and v as C is varied. While not as satisfactory, it is easier to show the existence of at least one point contained in a period-3 orbit by using basic calculus techniques to show the existence of a fixed point of f 3 different from the fixed points of f .…”

Damped-driven system of bouncing droplets leading to deterministically diffusive behavior

“…Finally, if we drive the ODE with some non-autonomous forcing f (t), the system exhibits even more complex behavior and can even become chaotic if it no longer satisfies the hypothesis of the Poincaré-Bendixson theorem [15,16,17]. Despite the simplicity of the concept, damped-driven systems, such as mode-locked lasers [7,18], rotation detonation engines [8], and chaotic logical circuits [19] can prove to be experimentally difficult, and even more so to analyze using dynamical systems techniques without significant model reduction.…”

“…While the trajectory-based formulations [44,45,46,47] are not able to find this analog, the energy gain-loss formulation [48] is able to find it, which reveals a fundamental connection between the dynamics in the macroscale phenomenon of the droplet and the nanoscale phenomenon of lasers that relies heavily on quantum effects. On the other end of the scale spectrum, rotation detonation engines [8,64] and electronic circuits [65,66] also have the same type of destabilization [67,19,68] . There is a plethora of other phenomena that seem to experience similar bifurcations, such as biolocomotion, porpoising in Formula 1, flutter of airplane wings, which have yet to be rigorously analyzed.…”

Birhythmic Analog Circuit Maze: A Nonlinear Neurostimulation Testbed

“…Clinically, this platform can serve as an initial stepping stone, prior to the use of animal models and human patients, in demonstrating the successful use of a stimulation regime under our hypothesis for a given application. As is a common method for realizing a dynamical system physically, an analog electronic circuit exhibiting the desired dynamics is constructed [ 30 , 31 , 32 ]. Due to the nature of widely used electrical stimulation for neurological implants, this medium will serve well as a testbed.…”