## 4 Numerical Example  
In this section, we present some numerical simulations that verify and illustrate the effectiveness of the theoretical analysis in Section 3. It is noted that the Adams-Bashforth-Moulton method with the step size 0.001 [25; 26]. We consider as master the fractional order permanent magnet synchronous motor (PMSM) model presented in [27]. A PMSM is a type of alternating current (AC) synchronous motor that uses permanent magnets to produce torque even at zero speed and can have a higher torque density compared to other types of motors. In terms of its mathematical model, it is considered a nonlinear coupling system with multiple variables. Traditionally, the PMSM was modeled as an integer-order dynamical system, which only takes into consideration the local knowledge of the states and inputs. The basic idea behind the fractional model in [27] is that, unlike integer calculus, fractional calculus involves an infinite number of terms. These terms can be employed to represent the past history of an arbitrary dynamical system. The considered model is of the form  
\[D_{t}^{p}X(t)=f(X(t)), \tag{29}\]  
where \(X(t)=\left(x_{1},x_{2},x_{3}\right)^{T}\),\[f\left(X(t)\right)=\left( \begin{array}{c}-x_{1}+x_{2}x_{3}\\ -x_{2}-x_{1}x_{3}+ax_{3}\\ b\left(x_{2}-x_{3}\right)\end{array} \right),\]  
\((a,b)=(100,10)\), and \(p=0.95\). As for the slave system, let us also consider the 4-component hyperchaotic fractional order system proposed in [28] with the addition of a control term yielding  
\[D_{t}^{q}Y(t)=BY(t)+g(Y(t))+U, \tag{30}\]  
with  
\[B=\left( \begin{array}{cccc}-10&10&0&0\\ 28&1&0&-1\\ 0&0&-\frac{8}{3}&0\\ 0&0&0&0\end{array} \right),\ G=\left( \begin{array}{c}0\\ -y_{1}y_{3}\\ -y_{1}y_{2}\\ 0.1y_{2}y_{3}\end{array} \right),\ \text{and}\ U=\left( \begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\\ u_{4}\end{array} \right).\]  
The authors of [28] showed that, subject to \(0.916\leq q\leq 1\), the system always exhibits hyperchaotic behavior. Hence, we choose the arbitrary fractional order \(q=0.94\). System (30) was proposed as the fractional counterpart of the integer-order hyperchaotic system proposed in [29]. A new modified generalized projective synchronization scheme was developed for (30) and applied to a secure communication system in [28]. The chaotic attractors of the proposed master and slave systems are depicted in Figures 1 and 2, respectively. The \(\mathbf{F}\)-\(\mathbf{M}\) synchronization strategy aims to force the error  
\[e\left(t\right)=\mathbf{F}\left(y_{1},y_{2},y_{3},y_{4}\right)-\mathbf{M} \left(t\right)\times\left(x_{1},x_{2},x_{3}\right)^{T} \tag{31}\]  
to zero as \(t\rightarrow+\infty\). Recall from Remark 1 that our choice of matrix \(\mathbf{M}\) and function \(\mathbf{F}\) can lead to different types of conventional synchronization schemes. In the following, we present numerical results confirming the validity and convergence of the proposed control schemes for \(d=m=4\), \(d=3<m,d=2<m\), respectively. Figure 1: Chaotic attractors of the master system (29) when \((a,b)=(100,10)\) and \(p=0.95\). **Case 1**. : _For \(d=m=4\), we may choose_  
\[\mathbf{F}=\left( \begin{array}{c}y_{1}+y_{3}+y_{4}\\ 2y_{2}+y_{4}\\ 3y_{3}\\ 4y_{4}\end{array} \right)\text{ and }\mathbf{M}\left(t\right)=\left( \begin{array}{ cccc}\frac{1}{t+1}&0&1\\ 1&\cos t&4\\ 1&2&0\\ exp\left(-t\right)&2&3\end{array} \right), \tag{32}\]  
_leading to_  
\[\mathbf{DF}\left(Y\left(t\right)\right)=\left( \begin{array}{ cccc}1&0&1&1\\ 0&2&0&1\\ 0&0&3&0\\ 0&0&0&4\end{array} \right)\text{ and }\mathbf{D}^{-1}=\left( \begin{array}{ cccc}1&0&0&0\\ 0&2&0&0\\ 0&0&3&0\\ 0&0&0&4\end{array} \right).\]  
_Our choice of control matrix \(C\) must satisfy condition (13) as stated in Theorem 1. For instance, we may consider_  
\[C=\left( \begin{array}{cccc}0&10&0&0\\ 28&2&0&-1\\ 0&0&0&0\\ 0&0&0&1\end{array} \right), \tag{33}\]  
_as all the eigenvalues of \(B-C\) have negative real parts. The resulting error system may be described by_  
\[\left\{