**5. Synchronization between Fractional-Order Glucose Insulin Systems**

Synchronization is a nonlinear phenomenon that was observed in biological systems; it is seen on isolated cells [15], clusters of cells as in organisms, and even in collective dynamics of populations [25]. Regarding the glucose-insulin system, it has been shown pancreatic *β*-cells also present a collective behavior whose synchronization underlies a small-world functional organization [24–26]. Thus, the synchronization is crucial to effectuate a pulsatile insulin liberation in cells, which guarantees more substantial hypoglycemic effects. Hence, we study the synchronization between fractional-order glucose-insulin regulatory systems. We expect that the synchronization state converges into a periodic behavior, because it is the typical response in a suband blood glucose concentrations. We define thject with normal metabolic conditions, allowing with this, the synchronization between the insulin e drive and response system, as follows

$$\begin{cases} D^{\theta\_1} \mathbf{x}\_1 &=& -a\_1 \mathbf{x}\_1 + 0.1 \mathbf{x}\_1 \mathbf{y}\_1 + 1.09 \mathbf{y}\_1^2 - 1.08 \mathbf{y}\_1^3 + 0.03 \mathbf{z}\_1 - 0.06 \mathbf{z}\_1^2 + a\_7 \mathbf{z}\_1^3 - 0.19, \\ D^{\theta\_2} \mathbf{y}\_1 &=& -a\_8 \mathbf{x}\_1 \mathbf{y}\_1 + 3.84 \mathbf{x}\_1^2 + 1.2 \mathbf{x}\_1^3 + 0.3 \mathbf{y}\_1 (1 - \mathbf{y}\_1) - 1.37 \mathbf{z}\_1 + 0.3 \mathbf{z}\_1^2 - 0.22 \mathbf{z}\_1^3 - 0.56, \\ D^{\theta\_3} \mathbf{z}\_1 &=& a\_{15} \mathbf{y}\_1 - 1.35 \mathbf{y}\_1^2 + 0.5 \mathbf{y}\_1^3 + 0.42 \mathbf{z}\_1 + 0.15 \mathbf{y}\_1 \mathbf{z}\_1. \end{cases} \tag{29}$$

and

$$\begin{array}{rcl}D^{\mathsf{fl}}\mathbf{x}\_{2} &=& -\mathfrak{d}\_{1}\mathbf{x}\_{2} + 0.1\mathfrak{x}\_{2}\mathfrak{y}\_{2} + 1.09\mathfrak{y}\_{2}^{2} - 1.08\mathfrak{y}\_{2}^{3} + 0.03\mathfrak{z}\_{2} - 0.06\mathfrak{z}\_{2}^{2} + \mathfrak{d}\mathfrak{y}\_{2}^{3} - 0.19 + \mathfrak{u}\_{1}, &\\D^{\mathsf{fl}}\mathfrak{y}\_{2} &=& -\hat{\mathfrak{a}}\mathfrak{s}\mathfrak{x}\mathfrak{y}\_{2} + 3.84\mathfrak{z}\_{2}^{2} + 1.2\mathfrak{z}\_{2}^{3} + 0.3\mathfrak{y}\_{2}(1 - \mathfrak{y}\_{2}) - 1.37\mathfrak{z}\_{2} + 0.3\mathfrak{z}\_{2}^{2} - 0.22\mathfrak{z}\_{2}^{3} - 0.56 + \mathfrak{u}\_{2}, &\\D^{\mathsf{fl}}\mathfrak{z}\_{2} &=& \mathfrak{d}\_{15}\mathfrak{y}\_{2} - 1.35\mathfrak{y}\_{2}^{2} + 0.5\mathfrak{y}\_{2}^{3} + 0.42\mathfrak{z}\_{2} + 0.15\mathfrak{y}\_{2}\mathfrak{z}\_{2} + \mathfrak{u}\_{3}. \end{array}$$

where *u*1, *u*2, *u*<sup>3</sup> in (30) represents the unknown control terms, and the error can be defined by

$$\begin{aligned} c\_1 &= x\_2 - x\_{1\prime} \\ c\_2 &= y\_2 - y\_{1\prime} \\ c\_3 &= z\_2 - z\_1 \end{aligned} \tag{31}$$

To achieve the synchronization, it is essential that the errors *ei* → 0 as *t* → ∞ with *i* = 1, 2, 3. Equation (31), together with (29) and (30), yield the error system

$$\begin{aligned} D^{\theta\_1}e\_1 &= -\mathfrak{A}\_1 \mathbf{x}\_2 + 0.1 \mathbf{x}\_2 \mathbf{y}\_2 + 1.09 \mathbf{y}\_2^2 - 1.08 \mathbf{y}\_2^3 + 0.03 \mathbf{z}\_2 - 0.06 \mathbf{z}\_2^2 + \mathfrak{h} \mathbf{y}\_2^3 + \\ &+ a\_1 \mathbf{x}\_1 - 0.1 \mathbf{x}\_1 \mathbf{y}\_1 - 1.09 \mathbf{y}\_1^2 + 1.08 \mathbf{y}\_1^3 - 0.03 \mathbf{z}\_1 + 0.06 \mathbf{z}\_1^2 - \mathfrak{a} \mathbf{y}\_1^3 + \mathfrak{u}\_1, \\ D^{\theta\_2} \mathbf{c}\_2 &= -\mathfrak{A}\_8 \mathbf{x}\_2 \mathbf{y}\_2 + 3.84 \mathbf{x}\_2^2 + 1.2 \mathbf{x}\_2^3 + 0.3 y\_2 (1 - y\_2) - 1.37 \mathbf{z}\_2 + 0.3 \mathbf{z}\_2^2 - 0.22 \mathbf{z}\_2^3 \\ &+ a\_8 \mathbf{x}\_1 \mathbf{y}\_1 - 3.84 \mathbf{x}\_1^2 - 1.2 \mathbf{x}\_1^3 - 0.3 y\_1 (1 - y\_1) + 1.37 \mathbf{z}\_1 - 0.3 \mathbf{z}\_1^2 + 0.22 \mathbf{z}\_1^3 + \mathfrak{u}\_2, \\ D^{\theta\_3} \mathbf{c}\_3 &= \hat{\mathfrak{a}}\_{15} \mathbf{y}\_2 - 1.35 y\_2^2 + 0.5 y\_2^3 + 0.42 \mathbf{z}\_2 + 0.15 y\_2 \mathbf{z}\_2 \\ &- a\_{15} \mathbf{y}\_1 + 1.35 y\_1^2 - 0.5 y\_1^3 - 0.42 \mathbf{z}\_1 - 0.15 y\_1 \mathbf{z}\_1 + u\_3. \end{aligned}$$

Let us define the active control functions *ui* with *i* = 1, 2, 3

$$\begin{array}{rcl} u\_1 &=& V\_1 + \mathfrak{h}\_1 \mathbf{x}\_2 - 0.1 \mathbf{x}\_2 \mathbf{y}\_2 - 1.09 \mathbf{y}\_2^2 + 1.08 \mathbf{y}\_2^3 - 0.03 \mathbf{z}\_2 + 0.06 \mathbf{z}\_2^2 - \mathfrak{h} \mathbf{y}\_2^3 - \\ & a\_1 \mathbf{x}\_1 + 0.1 \mathbf{x}\_1 \mathbf{y}\_1 + 1.09 \mathbf{y}\_1^2 - 1.08 \mathbf{y}\_1^3 + 0.03 \mathbf{z}\_1 - 0.06 \mathbf{z}\_1^2 + a\_7 \mathbf{z}\_1^3, \\ u\_2 &=& V\_2 + \mathfrak{h}\_8 \mathbf{x}\_2 \mathbf{y}\_2 - 3.84 \mathbf{x}\_2^2 - 1.2 \mathbf{x}\_2^3 - 0.3 y\_2 (1 - y\_2) + 1.37 \mathbf{z}\_2 - 0.3 \mathbf{z}\_2^2 + 0.22 \mathbf{z}\_2^3 \\ & - a\_8 \mathbf{x}\_1 \mathbf{y}\_1 + 3.84 \mathbf{x}\_1^2 + 1.2 \mathbf{x}\_1^3 + 0.3 y\_1 (1 - y\_1) - 1.37 \mathbf{z}\_1 + 0.3 \mathbf{z}\_1^2 - 0.22 \mathbf{z}\_1^3, \\ u\_3 &=& V\_3 - \mathfrak{h}\_{15} \mathbf{y}\_2 + 1.35 \mathbf{y}\_2^2 - 0.5 \mathbf{y}\_2^3 - 0.4 \mathbf{z}\_2 - 0.15 \mathbf{y}\_2 \mathbf{z}\_2 \\ & + a\_{15} \mathbf{y}\_1 - 1.35 \mathbf{y}\_1^2 + 0.5 \mathbf{y}\_1^3 + 0.4 2 \mathbf{z}\_1 + 0.15 \mathbf{y}\_1 \mathbf{z}\_1, \end{array} \tag{33}$$

where the linear functions *V*1, *V*2, *V*<sup>3</sup> are given by

$$\begin{aligned} V\_1 &= -c\_1, \\ V\_2 &= -c\_2, \\ V\_3 &= -c\_3. \end{aligned} \tag{34}$$

By using (33) and (34), the error system (32) becomes

⎡ ⎢ ⎣

$$
\begin{bmatrix} D^{q\_1} e\_1 \\ D^{q\_2} e\_2 \\ D^{q\_3} e\_3 \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} e\_1 \\ e\_2 \\ e\_3 \end{bmatrix} \,. \tag{35}
$$

The synchronization error vanishes eventually because of the eigenvalues are −1, −1, −1 in Equation (35).

The synchronization scenario is as follows. The drive system has a periodic behavior, while the response system is in a chaotic state. We study the Type-1 Diabetes Mellitus (parameter *a*8), since it is the most common disorder, and affects most world population as well as it is correlated with obesity. For this case, *a*<sup>8</sup> = 0.5 and *a*ˆ8 = 0.27 for drive and response systems, respectively, while *a*<sup>1</sup> = *a*ˆ1, *a*<sup>7</sup> = *a*ˆ7, and *a*<sup>15</sup> = *a*ˆ15. The fractional-order are *q*<sup>1</sup> = *q*<sup>2</sup> = *q*<sup>3</sup> = *q* = 0.95 in both systems with *x*1(0) = 0.53, *y*1(0) = 1.31, *z*1(0) = 1.03 and *x*2(0) = 0.5, *y*2(0) = 1.1, *z*2(0) = 1.3, for drive and response systems, respectively. Figure 10a–c show the phase planes between the periodic (free of T1DM) and chaotic (with T1DM) systems. Additionally, the synchronization error by considering (36) is given in Figure 11. Due to the error tends to zero as time evolves, we infer that the proposed control strategy is suitable for forcing the system with the disorder to a state free of T1DM. It is worth noting that the control strategy can be extended to incorporate uncertainties and improve the robustness of the synchronization using other approaches, as shown in [22,85]. From a practical biological point of view, for instance, recent works have employed optical-based control using a light-activated Na+ channel, to attain insulin from *β*-cells both in-vitro and vivo [86,87]. Therefore, our results could be useful for future works where the glucose-insulin system could be controlled with an artificial control signal.

**Figure 10.** Synchronization planes for the fractional-order glucose-insulin systems (29) and (30) with *a*<sup>8</sup> = 0.5, *a*ˆ8 = 0.27, and *q* = 0.95, respectively. (**a**) *x*<sup>1</sup> − *x*<sup>2</sup> phase plane, (**b**) *y*<sup>1</sup> − *y*<sup>2</sup> phase plane, (**c**) *z*<sup>1</sup> − *z*<sup>2</sup> phase plane.

**Figure 11.** Synchronization error between the fractional-order glucose-insulin systems (29) and (30) when *a*<sup>8</sup> = 0.5, *a*ˆ8 = 0.27, *q* = 0.95, and (*x*1, *y*1, *z*1, *x*2, *y*2, *z*2)=(0.53, 1.31, 1.03, 0.49, 1, 0.8), respectively.
