*4.3. Type-2 Diabetes Mellitus: Parameter a*<sup>7</sup> *as a Function of Fractional-Order q*

The abnormal insulin secretion of the pancreatic *β*-cells is commonly related to T2DM or non-insulin-dependent diabetes mellitus, which is known to be a disorder with insulin resistance [80–82]. The interconnection among T2DM, insulin resistance, and obesity relies on the *β*-cell dysfunction [80,83]. T2DM condition is characterized by the parameter *a*<sup>8</sup> in (5). Figure 8a shows the bifurcation diagram for the parameter *a*<sup>8</sup> with a fractional-order *q* = 0.98. The parameter *a*<sup>8</sup> is appropriate to understand the insulin resistance of the human body since it describes the effect of emitted insulin on glucose level [46]. In the bifurcation diagram, that phenomenon is detected when *a*<sup>8</sup> < 0.37, which is associated with chaotic behavior, as demonstrated by Proposition 2.

**Figure 8.** (**a**) Bifurcation diagram varying the T2DM parameter *a*<sup>8</sup> and setting *q* = 0.98 and (**b**) Two-dimensional bifurcation diagram for *a*<sup>8</sup> and fractional-order *q*, the chaotic behavior is denoted by red regions, and the periodic behavior (healthy behavior) is given by the blue regions.

**Proposition 2.** *When q*<sup>1</sup> = *q*<sup>2</sup> = *q*<sup>3</sup> ≡ *q* = 0.98*, and a*<sup>1</sup> = 2.04*, a*<sup>7</sup> = 2.01*, a*<sup>8</sup> = 0.27*, a*<sup>15</sup> = 0.3*, the system* (5) *exhibits a chaotic attractor.*

**Proof.** By applying Theorem 3, we can determine the instability measure *ρ*. When *ρ* is strictly positive, a chaos condition could be established. By selecting *q* = 0.98, *a*<sup>8</sup> = 0.27, and *w* = 100 the characteristic polynomial at *E*<sup>1</sup> = (0.814, 1.813, 1.320) is

$$
\lambda^{294} + 2.174\lambda^{196} + 54.782\lambda^{98} - 82.2,\tag{25}
$$

with unstable root *λ* = 1.0033, while at the equilibrium point *E*<sup>2</sup> = (0.63, 0.937, 0.879) is

$$
\lambda^{294} + 1.818\lambda^{196} + 2.903\lambda^{98} + 16.537,\tag{26}
$$

with unstable roots *λ*1,2 = 1.0089 ± 0.0140*i*, where the instability measure of the system is *ρ* = (*π*/2*w*) − 0.0138 > 0. Thus, the system (5) fulfills the essential requirement for getting chaos when *q* = 0.98 and *a*<sup>1</sup> = 2.04, *a*<sup>7</sup> = 2.01, *a*<sup>8</sup> = 0.27, *a*<sup>15</sup> = 0.3.

Additionally, we compute the Lyapunov exponents *λ*<sup>1</sup> = 0.56, *λ*<sup>2</sup> = 0, *λ*<sup>3</sup> = −21.03. Figure 8b sketches the two-dimensional bifurcation diagram for the T2DM parameter *a*<sup>8</sup> and the fractional-order *q*. Analogous previous cases, the red areas evolve to chaos, whereas the blue regions converge to a stable behavior (healthy condition). There is a linear fit between the fractional-order and T2DB. The lower the fractional-order, the lower the value for *a*8, where the T2DM disorder is observed. Besides, for *q* < 0.97 the T2DM disappear for 0.25 < *a*<sup>8</sup> < 0.7. These results suggest that the T2DM is not presented in the glucose-insulin system (5) for lowers fractional-orders.
