**6. Physical Realization of the Fractional-Order Glucose-Insulin System Based on an ARM Processor**

As well known, the experimental realization of fractional-order dynamical systems is a hot topic that has been attracting the attention of researchers since it is a path for demonstrating the complex dynamics, including chaos [35–39,44,45]. For fractional-order systems, there three typical approaches for getting electronic circuits: frequency-domain approximation, numerical algorithms, and the Adomian decomposition method [35–39,44,45]. The first-mentioned is not recommended for chaos detection, since it may induce incorrect results [62,63]. On the other hand, the second and latter approaches are good options for physically implementing fractional-order systems in re-programmable digital hardware [44,45]. Therefore, we chose the numerical algorithm approach for programming the ABM method. Subsequently, we select herein the ARM SoC Broadcom BCM2837B0 for the experimental verification of the fractional-order glucose-insulin regulatory system. The SoC contains an ARM core with 64-bit. An SDRAM LPDDR2 with 1GB. The ARM cores are capable of running at up to 1.4 GHz. It's possible to create an interface by using the GPIO port with a 16-bit monotonic voltage output D/A converter AD569. Figure 12a,b present the block diagram of the working principle and the main instructions of the pseudo-code, respectively.

After initializing the ARM processor, we set the initial values, *h*, *q*, *x*0, *y*0, *z*0. Because of the negative values of the system (1), a positive integer *φ* is needed to offset the time-series of the state-variables to avoid losing data. In this manner, all computed data are now positive. Next, we multiply the data by a positive integer *γ* to fit them to the DAC resolution of 214 bits. Finally, the obtained results visualize in an oscilloscope, as shown in Figures 13 and 14. We analyze the scenario related to hypoglycemia. First, we implement the case where system (1) presents the hypoglycemia condition, as given in Figure 13. As can be seen, the experimental phase portraits are pretty similar to those that are shown in Figure 1. Finally, the case when the fractional-order system (1) is free of hypoglycemia, i.e., an steady-state and, therefore, convergers to a periodic attractor, is given in Figure 14. Similar to the previous case, the experimental results are in good agreement with Figure 4. Subsequently, it indicates that the fractional-order glucose-insulin regulatory system was successfully realized on an ARM digital platform.

**Figure 12.** (**a**) Simplified diagram of the implementation of the fractional-order glucose-insulin regulatory system (1), and (**b**) the main steps of the proposed algorithm for implementing it on an ARM digital platform.

**Figure 13.** Experimental phase portraits of the fractional-order glucose-insulin regulatory system (1) showing hypoglycemia (chaos behavior) with *h* = 0.01, *a*<sup>1</sup> = 1.3, *a*<sup>7</sup> = 2.01, *a*<sup>8</sup> = 0.22, *a*<sup>15</sup> = 0.3, *q*<sup>1</sup> = *q*<sup>2</sup> = *q*<sup>3</sup> = 0.9 and (*x*0, *y*0, *z*0)=(0.5, 1.2, 1). (**a**) *x* − *y* plane, (**b**) *x* − *z* plane, (**c**) *y* − *z* plane.

**Figure 14.** Experimental phase portraits of the fractional-order glucose-insulin regulatory system (1) depicting a steady-state free of hypoglycemia with *h* = 0.01, *a*<sup>1</sup> = 1.55, *a*<sup>7</sup> = 2.01, *a*<sup>8</sup> = 0.22, *a*<sup>15</sup> = 0.3, *q*<sup>1</sup> = *q*<sup>2</sup> = *q*<sup>3</sup> = 0.9 and (*x*0, *y*0, *z*0)=(0.5, 1.2, 1).(**a**) *x* − *y* plane, (**b**) *x* − *z* plane, (**c**) *y* − *z* plane.
