**6. Conclusions**

Two autonomous Boolean networks that generate Boolean chaos have been introduced. From the mathematical model, it was shown that the logical states would never reach a fixed-point, and therefore, will oscillate permanently. The proposed Boolean chaos oscillators exhibited no dependence on incommensurate time-delays, as demonstrated by computing the Lyapunov exponents under various scenarios for the delay paths. The correct physical implementations of the two Boolean chaos oscillators are good evidence of the predicted conditions. Therefore, the BCOs are reliable and robust to be implemented with multiple circuit implementations, both discrete as integrated.

In particular, the synthesis of the chaotic oscillators in an integrated circuit has shown the benefits of a compact CMOS chaos generator with areas 0.0045 mm2 and 0.000 832 mm<sup>2</sup> for BCO-1 and BCO-2, respectively, as well as high-speed chaotic oscillations with relevant amplitude content up to 200 MHz. Several dynamical analyses such as time-series, chaotic attractors, Poincaré maps, and Lyapunov exponents validated the experimental results.

In this manner, the proposed Boolean chaos oscillators could be useful for various engineering applications, for instance, random number generators, since the design is straightforward, robust, compact, and can be implemented in many options of hardware without needing special considerations.

**Author Contributions:** Conceptualization, J.M.M.-P. and T.G.-C.; methodology, J.M.M.-P. and V.R.G.-D.; software, J.M.M.-P., V.R.G.-D., G.d.L.F.-C.; validation, V.R.G.-D., T.G.-C. and G.d.L.F.-C.; formal analysis, J.M.M.-P.; investigation, J.M.M.-P., T.G.-C., and L.d.C.G.-P.; writing—original draft preparation, J.M.M.-P. and T.G.-C.; writing—review and editing, J.M.M.-P., V.R.G.-D., L.d.C.G.-P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** J.M. Munoz-Pacheco acknowledges the National Council of Science and Technology (CONACYT) MEXICO for the financial support through Project No. 258880 (Proyecto Apoyado por el Fondo Sectorial de Investigacion para la Educacion). The Authors also thank CONACYT for the Lab project INFRA-2013-1-205873. The authors thankfully acknowledge computer resources, technical advice, and support provided by Laboratorio Nacional de Supercómputo del Sureste de México(LNS), a member of the CONACYT national laboratories, with project No. 202001025C.

**Conflicts of Interest:** The authors declare no conflict of interest.
