*4.1. Hypoglycemia: Parameter a*<sup>1</sup> *as a Function of Fractional-Order q*

For patients with diabetes, hypoglycemia emerges when the reduction of blood glucose concentration reduces below 185 3.9 mmol/L (70 mg/dL) [68]. This is a critical condition, since hypoglycemia may lead to a life-alarming state. In Equation (5), this complication is analyzed when considering the parameter *a*<sup>1</sup> and the fractional-order *q*. It means that, if the rate of insulin decrease, which is represented by *a*<sup>1</sup> in the system (5), gets low, then the hypoglycemia phenomenon emerge. Therefore, we suppose that the underlying system converges into a chaotic behavior as shown in Figure 2.

Figure 3a exhibits the bifurcation diagram of system (5) with a fixed fractional-order and considering *a*<sup>1</sup> as a critical parameter. The bifurcation diagram was made when considering the following: when the state-variable *x* intersects the Poincaré plane provided by *x* − *px* = 0 with *px* = 0.5, the measure *r* = *y*<sup>2</sup> + *z*<sup>2</sup> is delineated. It can be observed that system is stable for values of parameter *a*<sup>1</sup> > 1.5 but as the parameter diminishes the behavior turns chaotic.

**Figure 3.** (**a**) Bifurcation diagram varying the hypoglycemia parameter *a*<sup>1</sup> and setting *q* = 0.9, and (**b**) its Lyapunov exponent spectrum when *a*<sup>1</sup> = 1.3.

Additionally, we observed that the fractional-order *q* produces a shift concerning the bifurcation diagram showed in Ref. [46]. This consideration exemplifies the importance of considering a fractional-order derivative in the dynamical system, i.e., when values lesser than *a*<sup>1</sup> = 2.3 are set in the integer-order system [46], chaotic behavior was observed; however, this limit is different for the fractional-order model (*a*<sup>1</sup> ≤ 1.45). It is at this moment when we could mention that a disorder appears. The numerical results of Lyapunov exponents denoted by *λ<sup>i</sup>* with *i* = 1, 2, 3 are shown in Figure 3b for *a*<sup>1</sup> = 1.3 and *q* = 0.9 by applying Wolf's algorithm [69]. The fractional-order glucose-insulin system is chaotic because of the exponents are *λ*<sup>1</sup> > 0, *λ*<sup>2</sup> = 0 and *λ*<sup>3</sup> < 0 with |*λ*1| < |*λ*<sup>2</sup> + *λ*3|. Those results imply that the system is sensitive to tiny variations of its initial conditions [70,71].

Besides, a two-dimensional bifurcation diagram between the hypoglycemia parameter *a*<sup>1</sup> and *q* is presented in Figure 4. The unbounded behavior is represented by green regions, whereas chaos regions are denoted by red color. The black regions lead to healthy behavior (free of hypoglycemia). We found the lower the fractional-order, the lower the effect of *a*1. The basin of attraction in the plane *x*(0) − *y*(0) for *z*(0) = 1, *q* = 0.9 and *a*<sup>1</sup> = 1.3, is plotted in Figure 4b, the yellow region stand for a chaotic attractor shown in Figure 2, whereas initial conditions from blue region converge into a unbounded behavior. Finally, Figure 5a–c and Figure 6, presents the phase portraits and Lyapunov exponents, respectively, of healthy behavior for system (5) obtaining a Lyapunov exponent with magnitude zero and two negatives.

**Figure 4.** (**a**) Two-dimensional bifurcation diagram for the hypoglycemia parameter *a*<sup>1</sup> and fractional-order *q*, where green region stands for unbounded behavior, red for chaotic behavior, and blue regions lead to healthy (periodic) behavior. (**b**) Basin of attraction on the *x*(0) − *y*(0) plane with *z*(0) = 1, *q* = 0.9, and *a*<sup>1</sup> = 1.3 showing the chaotic behavior. The initial conditions marked in the yellow color lead into a bounded chaotic attractor, whereas the initial conditions in blue region converge into unbounded behavior.

**Figure 5.** Stable behavior of the fractional-order system (5) considering *q* = 0.9, *a*<sup>1</sup> = 1.55, and initial conditions (*x*0, *y*0, *z*0)=(0.5, 1.2, 1), with a integration step-size *h* = 0.01. (**a**) *x* − *y* phase plane, (**b**) *x* − *z* phase plane, (**c**) *y* − *z* phase plane.
