The mathematics describing Autonomous Boolean Networks has shown that they could display aperiodic patterns if the Boolean functions have instantaneous response times, the link time-delays are incommensurate, and their nodes perform asymmetric Boolean operations, such as the combination of logic exclusive-OR (XOR) and XNOR functions.
This research introduces two new chaotic oscillators based on autonomous Boolean networks (ABN), preserving asymmetrical logic functions. That means that the ABNs require a combination of XOR-XNOR logic functions. We demonstrate analytically that the two ABNs do not have fixed points, and therefore, can evolve to Boolean chaos. Using the Lyapunov exponent’s method, we also prove the chaotic behavior, generated by the proposed chaotic oscillators, is insensitive to incommensurate time-delays paths.
As a result, they can be implemented using distinct electronic circuits. More specifically, logic-gates–, GAL–, and FPGA–based implementations verify the theoretical findings.
An integrated circuit using a CMOS 180nm fabrication technology is also presented to get a compact chaos oscillator with relatively high-frequency.
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). 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.
We note a possible benefit of using ABNs can be to get Boolean chaos oscillators with relatively high oscillation frequencies and small form factors since they depend on logic functions only. However, we also found that the proposed ABNs have high sensitivity to the time-delay among network nodes for generating chaos behavior. From a practical point of view, that condition is very complicated to satisfy since the time-delays are heavily related to the electronics technology chosen for implementation. Due to the electronic logic gates being heterogeneous, they do not have the same intrinsic time-delay.
For details consult:
Munoz-Pacheco, J. M., GarcÃa-Chávez, T., Gonzalez-Diaz, V. R., de La Fuente-Cortes, G., & del Carmen Gómez-Pavón, L. (2020). Two New Asymmetric Boolean Chaos Oscillators with No Dependence on Incommensurate Time-Delays and Their Circuit Implementation. Symmetry,12(4), 506.