*4.2. Hyperinsulinemia: Parameter a*<sup>7</sup> *as a Function of Fractional-Order q*

Hyperinsulinemia means the quantity of insulin in the blood is higher than normal levels. Hyperinsulinemia is most often caused by insulin resistance, both humans and animals [72]. A condition in which the body is not capable of acts in the right form to the effects of insulin. Consequently, in order to compensate the high blood glucose levels, the pancreatic *β*-cells irrigate more insulin [73–76]. Hyperinsulinemia condition is analyzed in the fractional-order glucose-insulin regulatory system (5) by the parameter *a*7. Figure 7a shows the bifurcation diagram for different values of *a*<sup>7</sup> as a function of fractional order *q*.

**Figure 6.** Lyapunov spectrum of (5): *a*<sup>1</sup> = 1.55 and *q* = 0.9.

**Figure 7.** (**a**) Bifurcation diagram varying the hyperinsulinemia parameter *a*<sup>7</sup> and setting *q* = 0.95 and (**b**) Two-dimensional bifurcation diagram for *a*<sup>7</sup> and fractional-order *q* where the unbounded behavior is represented by the green regions; chaotic behavior is denoted by red regions, and the healthy behavior (free of hyperinsulinemia) is given by the blue regions.

From Figure 7a, we can observe that the system is stable when the values for *a*<sup>7</sup> are small, which describes the increased rate of insulin. If *a*<sup>7</sup> increases, the system becomes in a chaotic manner, which can be proved by Proposition 1 as follows

**Proposition 1.** *When q*<sup>1</sup> = *q*<sup>2</sup> = *q*<sup>3</sup> ≡ *q* = 0.95 *and a*<sup>1</sup> = 2.04*, a*<sup>7</sup> = 2.4*, a*<sup>8</sup> = 0.22*, a*<sup>15</sup> = 0.3*, the system* (5) *exhibits a chaotic attractor.*

**Proof.** To demonstrate the nonlinear behavior (chaotic behavior) in (5), it is mandatory that the instability measure *ρ* defined in (12) is nonnegative. When considering *q* = 0.95, *a*<sup>7</sup> = 2.4, and *w* = 100, the characteristic equation at the equilibrium point *E*<sup>1</sup> = (0.802, 1.866, 1.273) is

$$
\lambda^{285} + 2.149\lambda^{190} + 58.565\lambda^{99} - 102.703,\tag{23}
$$

with a unstable root *λ* = 1.0049, whereas the characteristic polynomial at *E*<sup>2</sup> = (0.606, 0.889, 0.812) is

$$
\lambda^{285} + 1.764\lambda^{190} + 1.658\lambda^{95} + 17.26,\tag{24}
$$

with unstable roots *λ*1,2 = 1.0090 ± 0.0137*i*, then *ρ* = (*π*/2*w*) − 0.0136 > 0. This result implies system (5) could generate a chaotic attractor when *q* = 0.95 and *a*<sup>1</sup> = 2.04, *a*<sup>7</sup> = 2.4, *a*<sup>8</sup> = 0.22, *a*<sup>15</sup> = 0.3.

Besides, the phenomenon antimonotonicity is stated in Figure 7a, which refers to the creation of period orbits followed by their nullification with reverse bifurcation sequences [77]. This phenomenon is one of the most common paths to chaos [78,79]. Antimonotonicity was found in Equation (5) by sweeping *a*<sup>7</sup> in the interval 2.6 ≤ *a*<sup>7</sup> ≤ 3.2 with *q* = 0.95. Additionally, we obtain the Lyapunov exponents *λ*<sup>1</sup> = 1.6617, *λ*<sup>2</sup> =0, *λ*<sup>3</sup> =-24.7646) indicating chaos.

On the other hand, Figure 7b gives the two-dimensional bifurcation diagram between the hyperinsulinemia parameter *a*<sup>7</sup> and the fractional-order *q*. The unbounded behavior is represented by the green regions; chaotic behavior is denoted by red regions and the healthy behavior (free of hyperinsulinemia) is given by the blue regions. We found that hyperinsulinemia disorder depends on the value of fractional-order. For values *q* < 0.92, the hyperinsulinemia tends to periodic oscillations.
