**7. Conclusions**

The dynamical analysis, synchronization, and physical realization of a glucose-insulin regulatory system has been presented by using Caputo's non-local fractional-order operator. In particular, we studied four common disorders, such as T1DM, T2DM, Hypoglycemia, and Hyperinsulinemia. We found that the fractional-order system switches between a chaotic behavior (a health disorder) and a disorder-free state, not only for the values of systems parameters, but also as a function of the fractional-order, due it adds more degrees of freedom in the model.To understand that insight, we computed two-dimensional bifurcations diagrams, which demonstrated the importance of considering the fractional-order (memory index) for getting a higher approximation of the observed behavior because fractional-order systems describe the whole-time domain in the solution, while the integer-order model is related to the local properties. Additionally, a phenomenon of antimonotonicity was observed in the parameter related to the hyperinsulinemia case. Besides, by applying the straightforward active control method, we showed that stable behavior in the fractional glucose-insulin system under the T1DM condition could be attained when it synchronizes with a disorder-free system. We remark that the synchronization can be extended to the remaining conditions. Finally, the electronics approach-based validation of chaotic and periodic behaviors was shown using an ARM digital platform. The experimental observations were in good agreement with the theoretical findings.

In this manner, the system-of-a-chip circuit designs are the right candidate for exploiting the advantages of the fractional-order models due to their simplicity, programmable characteristics, and portability, therefore increasing the fractional-order-based oncoming applications. As future work, an analysis related to robustness of the synchronization scheme will be developed.

**Author Contributions:** Conceptualization, J.M.M.-P.; Formal analysis, J.M.M.-P. and E.Z.-S; Methodology, E.Z.-S; Supervision, C.P.-C.; Validation, C.P.-C.; Writing—original draft, E.Z.-S.; Writing—review and editing, J.M.M.-P. All authors have read and agreed to the published version of the manuscript.

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

**Acknowledgments:** The authors thankfully acknowledge the computer resources, technical expertise and support provided by the Laboratorio Nacional de Supercómputo del Sureste de México, CONACYT member of the network of national laboratories. J.M. Muñoz-Pacheco thanks CONACyT/MEXICO for the financial support to through Project No. 258880 (Proyecto Apoyado por el Fondo Sectorial de Investigación para la Educación) ; VIEP-BUAP [2020]; and Plan de Trabajo CA [BUAP-CA-276]. C. Posadas-Castillo acknowledges CONACYT/Mexico under Research Grant No. 166654, A1-5-31628 and "Facultad de Ingeniería Mecánica y Eléctrica" (FIME-UANL). E. Zambrano-Serrano acknowledges CONACYT/Mexico (350385) for the financial support to complete a postdoctoral visit to "Facultad de Ingeniería Mecánica y Eléctrica" at UANL.

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