**2.4. Numerical examples**

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128 Numerical Simulation - From Brain Imaging to Turbulent Flows

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Example 2.1. Consider a VOFDDE:

$$D\_{t}^{\alpha(t)}\mathcal{y}\left(t\right) = \frac{2\mathcal{y}(t-2)}{1+\mathcal{y}(t-2)} - \mathcal{y}\left(t\right),$$
 
$$\mathcal{y}\left(t\right) = 0.5, \qquad t \le 0 \tag{29}$$

**Figure 11.** The numerical behavior of system (29) and chaotic attractors at *α* = 1.

**Figure 12.** The numerical behavior of system (29) and chaotic attractors at *α*(*t*) = 0.97.

**Figure 13.** The numerical behavior of system (32) and chaotic attractors at *α*(*t*) = *0.99* – (*0.01/100*)*t*.

In **Figures 11** and **1**(**a**, **b**) show the solutions *y*(*t*) and *y*(*t* − 2) of the system (29), for α = 1 and h = 0.1, whereas **Figure 1(c)** shows phase portrait of the system, i.e., plot of y(t) versus y(t − 2) for the same value of α. In this figure, it observed that the system (29) shows aperiodic chaotic behavior. Moreover, **Figure 12** shows the plot of the numerical solutions *y*(*t*) and the plot of *y*(*t*) versus *y*(*t* − 2), respectively, at *α*(*t*) = 0.97. In the following, we choose different cases for α(t). **Figures 13** and **14** show the solutions y(t) and y(t − 2) of the given system for *a*(*t*) = 0.99 − (0.01/100)*t* and *α*(*t*) = 0.95 − (0.01/100)*t*, respectively. The chaotic portrait for these values of *α*(*t*) is shown. In **Figure 15**, we have decreased the value of *α*(*t*), where *α*(*t*) = 0.85 − (0.01/100)*t*, and observed that the system becomes periodic.

**Figure 14.** The numerical behavior of system (29) and chaotic attractors at *α*(*t*) = *0.95* – (*0.01/100*)*t*.

**Figure 11.** The numerical behavior of system (29) and chaotic attractors at *α* = 1.

130 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 12.** The numerical behavior of system (29) and chaotic attractors at *α*(*t*) = 0.97.

**Figure 13.** The numerical behavior of system (32) and chaotic attractors at *α*(*t*) = *0.99* – (*0.01/100*)*t*.

In **Figures 11** and **1**(**a**, **b**) show the solutions *y*(*t*) and *y*(*t* − 2) of the system (29), for α = 1 and h = 0.1, whereas **Figure 1(c)** shows phase portrait of the system, i.e., plot of y(t) versus y(t − 2) for the same value of α. In this figure, it observed that the system (29) shows aperiodic chaotic

**Figure 15.** The numerical behavior of system (29) and chaotic attractors at *α*(*t*) = *0.85* – (*0.01/100*)*t*.

#### **2.5. The four-year life cycle of a population of lemmings model**

Lemmings are small rodents, usually found in or near the Arctic, in tundra biomes. It used to be that every three to 4 years, there were massive numbers of lemmings in the mountains and then the next year it was gone. Therefore, it has been an interesting thing to try to find out why. Are there factors that affect the lemming population such as climate, temperature, precipitation, and the like, that is, what we refer to as the lemming cycle. The modern scientific study of lemmings started with work carried out by the Norwegian Professor of Zoology Robert Collett, who at the end of the nineteenth century gathered a great deal of information about lemmings. To try to understand why lemmings fluctuate both regularly and extensively is indeed an important problem in ecology [34]. In [27], Tavernini has solved a model of the 4 year life cycle of a population of lemmings in an integer order. In [25], Bhalekar and DaftardarGejji have studied the model in a fractional order. In the following example [27], we study the extension of the lemming model:

Example 2.2. Consider the variable-order version of the four-year life cycle of a population of lemmings

$$D\_r^{a(t)}y(t) = 3.5y(t)\left(1 - \frac{y\left(t - 0.74\right)}{19}\right), y(0) - 19.00001, \qquad y(t) = 19, \tag{30}$$

**Figure 16.** The numerical behavior of system (33) and plots of the (*t*), *y*(*t* – 0.74) relation for α = 1.

**Figure 17.** The numerical behavior of system (33), and plots of the *y*(*t*), *y*(*t* – 0.74) relation for *α*(*t*) = 0.97.

**Figure 18.** The numerical behavior of system (33) and the stretching phenomena for *α*(*t*) = 0.99 – (0.01/100)*t*.

**Figure 16** shows the solutions *y*(*t*) and *y*(*t* − 0.74) of system (33) for α = 1 and h = 0.1 and shows phase portrait of *y*(*t*) versus y(t − 0.74) for the same value of *α*. **Figure 17** shows the plot of the numerical solutions *y*(*t*) and the plot of *y*(*t*) versus *y*(*t* − 0.74), respectively, at (*t*) = 0.97. The numerical results for VOFDDEs at different values of *α*(*t*) are given in **Figures 18**–**22**. These figures show the stretching phenomena between the numbers of lemmings in y(t) versus y(t − 2), for the values *α*(*t*) = *0.99* − (*0.01*/*100*)*<sup>t</sup>* , *α*(*t*) = *0.98* − (*0.02*/*100*)*<sup>t</sup>* ; *α*(*t*) = *0.95* − (*0.01*/*100*)*<sup>t</sup>* , *α*(*t*) = *0.87* − (*0.02*/*100*)*<sup>t</sup>* , and *α*(*t*) = *0.85* − (*0.01*/*100*)*<sup>t</sup>* , respectively. It is observed that the phase portrait gets stretched as the value of α(t) decreases, and this stretching is towards positive side of the axis.

**Figure 19.** The stretching.

Gejji have studied the model in a fractional order. In the following example [27], we study the

Example 2.2. Consider the variable-order version of the four-year life cycle of a population of

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**Figure 16.** The numerical behavior of system (33) and plots of the (*t*), *y*(*t* – 0.74) relation for α = 1.

**Figure 17.** The numerical behavior of system (33), and plots of the *y*(*t*), *y*(*t* – 0.74) relation for *α*(*t*) = 0.97.

**Figure 18.** The numerical behavior of system (33) and the stretching phenomena for *α*(*t*) = 0.99 – (0.01/100)*t*.

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132 Numerical Simulation - From Brain Imaging to Turbulent Flows

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**Figure 20.** The stretching phenomena for *α*(*t*) = *0.98* − (*0.02/100*)*t*. Phenomena for *α*(*t*) = 0.95 − (0.01/100)*t*.

**Figure 21.** The stretching phenomena for α(t) = 0.87 − (0.02/100)t.

**Figure 22.** The stretching phenomena for α(t) = 0.85 − (0.01/100)t.

#### **2.6. The enzyme kinetics with an inhibitor molecule model**

An enzyme inhibitor is a molecule that binds to an enzyme and decreases its activity or completely inhibits the enzyme catalytic activity. It is well known that all these inhibitors follow the same rule to interplay in enzyme reaction. Furthermore, there are many factors that affect enzyme's activity, such as temperature and pH. Many drug molecules are enzyme inhibitors, so their discovery and improvement are an active area of research in biochemistry and pharmacology to protect enzyme from any change. Therefore, studying the enzyme kinetics and structure–function relationship is vital to understand the kinetics of enzyme inhibition that in turn is fundamental to the modern design of pharmaceuticals in industries [35]. The frequency conversion mechanism in enzymatic feedback systems has been investi‐ gated with computer simulations in 1984 by Okamoto and Hayashi [36]. In [25], Bhalekar and Daftardar-Gejji have been study the enzyme kinetics with an inhibitor molecule model in a fractional order. In the following, we will study the model in [25] in general form where the derivative is given in a VOF.

Example 2.3. Consider the variable-order version of four-dimensional enzyme kinetics with an inhibitor molecule

$$\begin{aligned} D\_{\boldsymbol{\gamma}}^{a(t)} \boldsymbol{\gamma}\_{1}(t) &= 10.5 - \frac{\boldsymbol{\gamma}\_{1}(t)}{1 + 0.0005 \boldsymbol{\gamma}\_{4}^{4}(t - 4)}, \\\\ D\_{\boldsymbol{\gamma}}^{a(t)} \boldsymbol{\gamma}\_{2}(t) &= \frac{\boldsymbol{\gamma}\_{1}(t)}{1 + 0.0005 \boldsymbol{\gamma}\_{4}^{4}(t - 4)} - \boldsymbol{\gamma}\_{2}(t), \\\\ D\_{\boldsymbol{\gamma}}^{a(t)} \boldsymbol{\gamma}\_{3}(t) &= \boldsymbol{\gamma}\_{2}(t) - \boldsymbol{\gamma}\_{3}(t), \\\\ D\_{\boldsymbol{\gamma}}^{a(t)} \boldsymbol{\gamma}\_{4}(t) &= \boldsymbol{\gamma}\_{3}(t) - 0.5 \boldsymbol{\gamma}\_{4}(t), \\\\ \boldsymbol{\gamma}(t) &= [60, 10, 10, 20]^{\boldsymbol{r}}, t \le 0. \end{aligned} \tag{31}$$
 
$$\begin{aligned} \boldsymbol{\gamma}(t) &= [60, 10, 10, 20]^{\boldsymbol{r}}, t \le 0. \end{aligned} \tag{32}$$

**Figure 23.** The numerical behavior of system (34) at *α* = 1 (left) and *α* = 95 (right).

**Figure 21.** The stretching phenomena for α(t) = 0.87 − (0.02/100)t.

134 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 22.** The stretching phenomena for α(t) = 0.85 − (0.01/100)t.

**2.6. The enzyme kinetics with an inhibitor molecule model**

An enzyme inhibitor is a molecule that binds to an enzyme and decreases its activity or completely inhibits the enzyme catalytic activity. It is well known that all these inhibitors follow the same rule to interplay in enzyme reaction. Furthermore, there are many factors that affect enzyme's activity, such as temperature and pH. Many drug molecules are enzyme inhibitors, so their discovery and improvement are an active area of research in biochemistry and pharmacology to protect enzyme from any change. Therefore, studying the enzyme kinetics and structure–function relationship is vital to understand the kinetics of enzyme

**Figure 23** shows the solutions *y*(*t*), (1≤ *i* ≤ 4), with step size h= *1*, for = *1*, and *α* = *95*, respectively. Whereas **Figures 24**–**26** show the numerical results in case of variable order at (*t*) = 0.99 − (0.01/100)*t, α*(*t*) = *0.95* − (*0.01*/*100*)*t, α*(*t*) = *0.91* − (*0.01/100*)*t*, and *α*(*t*) = *0.83* − (*0.01/100*)*t*, respectively. For 0.90 < α(t) ≤ 1, the height of oscillations of the solutions increases as t increases as shown in **Figures 24** and **25**, while in **Figure 26**, the system settles down for sufficiently large t, for *α*(*t*) < *90*.

**Figure 24.** The numerical behavior of system (34) at *α*(*t*) = 0.99 – (0.01/100)*t* (left) and *α*(*t*) = 0.95 – (0.01/100)*t* (right).

**Figure 25.** The numerical behavior of system (34) at *α*(*t*) = 0.91 – (0.01/100)*t*.

**Figure 26.** The numerical behavior of system (34) at *α*(*t*) = 0.83 – (0.01/100)*t*.
