Forced synchronization of autonomous dynamical Boolean networks

Abstract

We present the design of an autonomous time-delay Boolean network realized with readily available electronic components. Through simulations and experiments that account for the detailed nonlinear response of each circuit element, we demonstrate that a network with five Boolean nodes displays complex behavior. Furthermore, we show that the dynamics of two identical networks display near-instantaneous synchronization to a periodic state when forced by a common periodic Boolean signal. A theoretical analysis of the network reveals the conditions under which complex behavior is expected in an individual network and the occurrence of synchronization in the forced networks. This research will enable future experiments on autonomous time-delay networks using readily available electronic components with dynamics on a slow enough time-scale so that inexpensive data collection systems can faithfully record the dynamics.

Two New Asymmetric Boolean Chaos Oscillators with No Dependence on Incommensurate Time-Delays and Their Circuit Implementation

“…It means that the chaos should be generated consistently for a wide range of parameter values. In the ongoing literature, the reported autonomous Boolean networks only show chaos in certain ranges of the feedback delays [10,30,31,[35][36][37][38][39], as was discussed in the Introduction section. At the experimental level, those approaches incorporate an even number of logic NOT gates in the link to act as a time-delay buffer to add extra signal propagation times.…”

“…The signal propagation time from node j to node i is τ ij for i, j = a, b, c. Theorem 3. For an autonomous Boolean network given by the system of Equation 2, the orbits are always oscillating [36].…”

“…In the context of ABNs, deterministic chaos, also known as Boolean chaos, was initially demonstrated by using Boolean functions implemented with electronic logic circuits (logic gates and field-programmable gate arrays FPGAs) [10,[35][36][37][38]. At circuit level, the basic principle for obtaining Boolean chaos depends on three main characteristics; the asymmetry between the logic states, the short-pulse rejection phenomenon, and, most importantly, the degradation effect [30,31].…”

“…For an autonomous Boolean network given by the system of Equation (7), the orbits are always oscillating[36].…”

Pinning controllability of autonomous Boolean control networks

“…In [31], Ghil and Mullhaupt have introduced the Boolean delay equations as an autonomous BN (ABN), and this model has then been widely applied to yeast cell cycle and electronic circuits [32]. In [33], based on the logic gates, Rivera-Durón has built an electronic realization of an ABN of five nodes with external Boolean signal that can be regarded as control input, namely autonomous Boolean control network (ABCN) (see Example 3 for reference), and the forced synchronization issue of two identical ABNs (forced by a common external signal) has been addressed. The recent work in [34] has shown that the analysis on complex dynamics in an electronic circuit is beneficial for capturing the qualitative aspects of the structure and dynamics of GRNs.…”