**2.7. The Chen system model**

**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*.

136 Numerical Simulation - From Brain Imaging to Turbulent Flows

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

The atmosphere is a layer of gases surrounding the planet Earth that has, at each altitude above each point of the Earth's surface, a density, a pressure, a temperature, etc., and all these vary over time. It is of course unthinkable to know all this infinite number of data and to understand something about it. Approximations must be made. Edward Norton Lorenz (1917–2008) is a pioneer of chaos theory. He introduced the strange attractor notion and coined the term butterfly effect. In 1963 [37], he developed a simplified mathematical model for atmospheric convection when he studied the atmospheric convection. His model of the atmosphere was reduced to only three numbers x, y, and z and the evolution of the atmosphere to a tiny equation, where each point x, y, and z in space symbolizes a state of the atmosphere and the evolution follows a vector field. Later, several dynamical systems exhibiting chaos have been presented in various branches of science [24]. For example, in 1999, Chen and Ueta found another simple three-dimensional autonomous system, which is not topologically equivalent to Lorenz's system and which has a chaotic attractor [38]. In [39], Chen proved that "The Chen system is a special case of the Lorenz system." In 2004, Li and Peng [40] have presented Chen system with a fractional order. In 2012, Bhalekar et al. [41] proposed the fractional-order Chen system with time delay. In the following example, we will extend Chen system to a VOF, which can be more general.

Example 2.4. Consider the generalization of the delay fractional-order version of the Chen system [3] which involves the variable order:

$$D\_t^{\alpha(t)}\mathbf{x}(t) = a(\mathbf{y}(t) - \mathbf{x}(t-\tau)),$$

$$D\_{\iota}^{\alpha(t)}\varphi(t) = (c-a)\chi(t-\tau) - \chi(t)z(t) + c\chi(t),$$

$$D\_{\iota}^{\alpha(\iota)}z\left(t\right) = x\left(t\right)\left\chi\left(t\right) - bz\left(t-\tau\right)\_{\ast}\right.$$

$$\mathbf{x}(t) = 0.2, \mathbf{y}(t) = 0, \mathbf{z}(t) = 0.5 \\ \text{for } t \in [-\tau, 0], \tag{32}$$

on the interval [*0,30*] and with step size *h* = *0.001*.

At = *1*, **Figure 27** shows the numerical solution for *τ* = *0.005* and shows chaotic *yz* phase portrait for this case. **Figure 28** shows the numerical solution and chaotic *yz* phase portrait at *α* = *0.97* and *τ* = *0.005*. Moreover, for the variable order, we choose different cases for *α*(*t*):

At *α*(*t*) = *0.97* – (*0.01/100*)*t*, **Figure 29** shows the numerical solution *y*(*t*) and chaotic *yz* phase portrait of this example for *τ* = *0.005*. When we increase the value of τ to 0.015, the results of the numerical solution and chaotic *yz* phase portrait become stable as shown in **Figure 30**. At *α*(*t*) = *0.94* – (*0.01/100*)*t*, **Figure 31** shows the numerical solution for *τ* = *0009*, and a limit cycle phase portrait was observed. When we increase the value of τ to 0.011, the chaotic *yz* phase portrait becomes stable as shown in **Figure 32**. **Figure 33** shows the numerical solution for *τ* = *0.005* and chaotic *yz* phase portrait at *α*(*t*) = *89* − (*0.01/100*)*t*. It is observed that even in onedimensional delayed systems of variable order, chaotic behavior can be shown, and subjected to some critical order, the system changes its nature and becomes periodic. In some cases, it is observed that the phase portrait gets stretched as the order of the derivative is reduced. According to the numerical test examples, it can be concluded that the proposed numerical technique is a powerful technique to calculate approximate solution of VOFDDEs.

**Figure 27.** The numerical behavior and chaotic attractors for = *1*, and *τ* = *0.005*.

**Figure 28.** The numerical behavior and chaotic attractors for *α = 0.97* and *τ = 0.005*.

**Figure 29.** The numerical behavior and chaotic attractors for *α*(*t*) = 0.97 – (*0.01/100*)*t* and *τ = 0.005*.

**Figure 30.** The numerical behavior and chaotic attractors for *α*(*t*) = *0.97* – (*0.01*/*100*)*t* and *τ* = *0.015*.

portrait becomes stable as shown in **Figure 32**. **Figure 33** shows the numerical solution for *τ* = *0.005* and chaotic *yz* phase portrait at *α*(*t*) = *89* − (*0.01/100*)*t*. It is observed that even in onedimensional delayed systems of variable order, chaotic behavior can be shown, and subjected to some critical order, the system changes its nature and becomes periodic. In some cases, it is observed that the phase portrait gets stretched as the order of the derivative is reduced. According to the numerical test examples, it can be concluded that the proposed numerical

technique is a powerful technique to calculate approximate solution of VOFDDEs.

**Figure 27.** The numerical behavior and chaotic attractors for = *1*, and *τ* = *0.005*.

138 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 28.** The numerical behavior and chaotic attractors for *α = 0.97* and *τ = 0.005*.

**Figure 29.** The numerical behavior and chaotic attractors for *α*(*t*) = 0.97 – (*0.01/100*)*t* and *τ = 0.005*.

**Figure 31.** The numerical behavior and chaotic attractors for *α*(*t*) = *0.94* – (*0.01*/*100*)*t* and *τ = 0.009*.

**Figure 32.** The numerical behavior and chaotic attractors for *α*(*t*) = *0.94* – (*0.01*/*100*)*t* and *τ = 0.011*.

**Figure 33.** The numerical behavior and chaotic attractors for *α*(*t*) = *0.89* – (*0.01*/*100*)*t* and *τ = 0.005*.
