## 5 Simulation Results
### Example 2  
In this section, we extend the application of our proposed control approach to the FO Liu system. By incorporating time delays, we aim to further validate the effectiveness and applicability of our control methods across different chaotic systems. We begin by transforming the traditional FO Liu system into a form that incorporates multiple time delays [29], as represented in Equation (51). The system dynamics are governed by three state variables \(x_{1}\), \(x_{2}\), and \(x_{3}\), with corresponding fractional derivatives in the Caputo sense denoted by \(D_{t}^{v}\). The system coefficients \(a\), \(b\), \(c\), and \(d\) are predefined constants, while \(\tau=1\) represents the time delay. We select specific coefficient values (\(a=2.5\), \(b=-4\), \(c=-5\), \(d=4\)) and an initial condition of \(\left[\begin{array}{cccc}0.1&0.1&0.1\\ \end{array}\right]\) to simulate the FO Liu system without control. The phase plot of the system is presented in Figure 6.  
Figure 5: Comparison of states of FO and integer-order system modes presented in [46].  
Figure 6: Phase plot of the open-loop FO Liu system for \(v=0.95\).  
\[\left\{\begin{array}{l}D_{t}^{v}x_{1}=-x_{1}(t-\tau)+x_{2}^{2}(t)\\ D_{t}^{v}x_{2}=ax_{2}(t-\tau)+bx_{1}(t)x_{3}(t)\\ D_{t}^{v}x_{3}=cx_{3}(t-\tau)+dx_{1}(t)x_{2}(t)\end{array}\right. \tag{52}\]  
It presents the phase plot of the FO Liu system without control, showcasing the relationships between the state variables and visually representing the system's attractor. This allows us to identify any attractor structure and assess the system's stability and dynamic behavior. In [29], the system was studied without any limitations on controller.  
By contrasting the results obtained with and without control, we can evaluate the effectiveness of our methods in stabilizing the system and regulating its behavior.  
Without control, the time-delayed FO Liu system exhibits unstable states for every \(\tau\geq 0\). Thus, we introduce controllers as follows:  
\[\left\{\begin{array}{l}D_{t}^{v}x_{1}=-x_{1}(t-\tau)+x_{2}^{2}(t)+b_{1}^{{} \text{sat}}(u_{1}(t))\\ D_{t}^{v}x_{2}=ax_{2}(t-\tau)+bx_{1}(t)x_{3}(t)+b_{2}^{{}^{\prime}}{\text{sat }}(u_{2}(t))\\ D_{t}^{v}x_{3}=cx_{3}(t-\tau)+dx_{1}(t)x_{2}(t)+b_{3}^{{}\text{sat}}(u_{3}(t) )\end{array}\right. \tag{53}\]  
We set the input coefficients \(b_{1}^{{}^{\prime}}=b_{2}^{{}^{\prime}}=b_{3}^{{}^{\prime}}=1\), and the vector Lyapunov function is defined as follows:  
\[V(x_{i}(t))=\Sigma_{i=0}^{3}{x_{i}}^{2}=V_{1}+V_{2}+V_{3} \tag{54}\]  
Selecting \(\zeta=3/2\) satisfies the requirements of Theorem 4. We can derive the following inequality:  
\[\
abla_{x_{i}}V_{i}(x_{i})x_{i}(t-1)\leq 2\|x_{i}(t)\|\|x_{i}(t-1)\|\leq 3\|x_{i}( t)\|^{2}, \tag{55}\]  
\[\left\{\begin{array}{l}\
abla_{x_{1}}V_{1}(x_{1})g_{1}(x(t))\leq 2\|x_{1}(t) \|\|x_{2}(t)\|^{2}\\ \
abla_{x_{2}}V_{2}(x_{2})g_{2}(x(t))\leq 8\|x_{1}(t)\|\|x_{2}(t)\|\|x_{3}(t) \|\\ \
abla_{x_{3}}V_{3}(x_{3})g_{3}(x(t))\leq 8\|x_{1}(t)\|\|x_{2}(t)\|\|x_{3}(t) \|\end{array}\right. \tag{56}\]  
Choosing \(u_{1}(x(t))=9x_{1}(t)\), \(u_{2}(x(t))=15x_{2}(t)\), and \(u_{3}(x(t))=13x_{3}(t)\), and based on Theorem 4, the matrix \(W\) is defined as follows:  
\[W=\left[\begin{array}{ccc}10&0&0\\ 0&28.5&0\\ 0&0&24\end{array}\right].\]  
It is evident that the matrix \(W\) is an M-matrix. We now proceed to analyze the closed-loop system described by (53) with the proposed controller. The time responses of the controlled FO Liu system are illustrated in Figures 7 and 8, showcasing the system's behavior under the influence of the designed controller. These figures provide insights into the convergence speed and stability of the controlled system. Figure 9 depicts the output of the saturation control.