## 3 The Synchronization Condition for Exact LS between Systems (2) and (3)  
Condition (4) is a functional relation between \(y_{i}\) and \(x_{i}\), \(i=1,2,3,4\). Then, LS between systems (2) and (3) can be regarded as a special type of GS between them, which can be investigated by using the auxiliary system approach [28]. The auxiliary system corresponding to systems (2) and (3) is given by  
\[\begin{split}\dot{z}_{1}&=-az_{1}+az_{2}+z_{4}+k_{ 1}(x_{1}(t-\tau)-z_{1}),\\ \dot  
the synchronization error system for condition (6) can be given as  
\[\dot{e}_{1} =-(k_{1}+a)e_{1}+ae_{2}+e_{4}, \tag{7}\] \[\dot{e}_{2} =\alpha e_{1}+(c-k_{2})e_{2}-u_{0}e_{3}-e_{1}e_{7}-e_{3}e_{5},\] \[\dot{e}_{3} =v_{0}e_{1}+u_{0}e_{2}-(b+k_{3})e_{3}+e_{1}e_{6}+e_{2}e_{5},\] \[\dot{e}_{4} =w_{0}e_{2}+v_{0}e_{3}+(r-k_{4})e_{4}+e_{2}e_{7}+e_{3}e_{6}.\]  
Condition (6) becomes  
\[||e_{i}||\to 0,\quad i=1,2,3,4,\quad for\quad t\to\infty. \tag{8}\]  
Consider the Laplace transform, defined as follows:  
\[E_{i}(s) =L[e_{i}]=\int_{0}^{+\infty}e_{i}(t)e^{-st}dt, \tag{9}\] \[e_{i}(t) =L^{-1}[E_{i}]=\frac{1}{2\pi j}\int_{\sigma-j\infty}^{\sigma+j \infty}E_{i}(s)e^{st}ds,\quad i=1,2,3,4.\]  
By taking the Laplace transform on both sides of system (7), it yields  
\[(sI_{4}-M) \begin{bmatrix}E_{1}\\ E_{2}\\ E_{3}\\ E_{4}\end{bmatrix} = \begin{bmatrix}e_{10}\\ W_{1}+e_{20}\\ W_{2}+e_{30}\\ W_{3}+e_{40}\end{bmatrix}  \tag{10}\]  
where \(e_{0i}\), \(i=1,2,3,4\) are the given initial values for system (7), \(I_{4}\) is the \(4\times 4\) identity real matrix,  
\[M= \begin{bmatrix}-k_{1}-a&a&0&1\\ \alpha&c-k_{2}&-u_{0}&0\\ v_{0}&u_{0}&-b-k_{3}&0\\ 0&w_{0}&v_{0}&r-k_{4}\end{bmatrix}  \tag{11}\]  
and \(W_{i}\), \(i=1,2,3\) are the Laplace transform of the nonlinear parts in the last three equations in system (7), respectively,  
\[W_{1} =-\int_{0}^{+\infty}[e_{1}e_{7}+e_{3}e_{5}]e^{-st}dt,\] \[W_{2} =\int_{0}^{+\infty}[e_{1}e_{6}+e_{2}e_{5}]e^{-st}dt,\] \[W_{3} =\int_{0}^{+\infty}[e_{2}e_{7}+e_{3}e_{6}]e^{-st}dt.\]  
Solving Equation (10) by using Cramer's rule, one has  
\[E_{1} =\frac{1}{D(s)} \begin{vmatrix}e_{10}&a&0&1\\ W_{1}+e_{20}&c-k_{2}&-u_{0}&0\\ W_{2}+e_{30}&u_{0}&-b-k_{3}&0\\ W_{3}+e_{40}&w_{0}&v_{0}&r-k_{4}\end{vmatrix} ,\] \[E_{2} =\frac{1}{D(s)} \begin{vmatrix}-k_{1}-a&e_{10}&0&1\\ \alpha&W_{1}+e_{20}&-u_{0}&0\\ v_{0}&W_{2}+e_{30}&-b-k_{3}&0\\ 0&W_{3}+e_{40}&v_{0}&r-k_{4}\end{vmatrix} ,\] \[E_{3} =\frac{1}{D(s)} \begin{vmatrix}-k_{1}-a&a&e_{10}&1\\ \alpha&c-k_{2}&W_{1}+e_{20}&0\\ v_{0}&u_{0}&W_{2}+e_{30}&0\\ w_{0}&W_{3}+e_{40}&r-k_{4}\end{vmatrix} ,\]\[E_{4}=\frac{1}{D(s)} \begin{vmatrix}-k_{1}-a&a&0&e_{10}\\ \alpha&c-k_{2}&-u_{0}&W_{1}+e_{20}\\ v_{0}&u_{0}&-b-k_{3}&W_{2}+e_{30}\\ 0&w_{0}&v_{0}&W_{3}+e_{40}\end{vmatrix} , \tag{12}\]  
where  
\[D(s)=s^{4}+\beta_{1}s^{3}+\beta_{2}s^{2}+\beta_{3}s+\beta_{4}, \tag{13}\]  
\[\beta_{1} =a+b-c-k-r,\] \[\beta_{2} =(b-\alpha-c-r)a-(c+k+r)b+(k+r)c+kr+{u_{0}}^{2},\] \[\beta_{3} =(a-k-r){u_{0}}^{2}+{u_{0}}\,v_{0}+((\alpha+c)r-(\alpha+c+r)b)a+( ck+rc+kr)b\] \[\quad-krc-\alpha\,w_{0}-{v_{0}}^{2},\] \[\beta_{4} =r(k-a){u_{0}}^{2}+(w_{0}-ar-\alpha){u_{0}}\,v_{0}+{c{v_{0}}}^{2} +(\alpha+c)abr-(krc+\alpha\,w_{0})b.\]  
\(D(s)\) is just the characteristic polynomial of matrix (11). In fact, we do not need to know the actual formulas of the numerators of \(E_{i}\), \(i=1,2,3,4\). They must have the following forms:  
\[E_{i}=\frac{h_{i0}(s)}{D(s)}+\sum_{j=1}^{3}\frac{h_{ij}(s)}{D(s)}W_{j},\quad i =1,2,3,4, \tag{14}\]  
in which \(h_{ij}(s)\) (\(i=1,2,3,4\), \(j=0,1,2,3\)) are polynomials of \(s\) whose orders are less than four. Taking the inverse Laplace transform for \(E_{i}\), \(i=1,2,3,4\) and considering the convolution theorem in the Laplace transform produces  
\[ \begin{split} e_{i}&=\gamma_{i0}(t)-\int_{0}^{t} \gamma_{i1}(t-\tau)[e_{1}e_{7}+e_{3}e_{5}]d\tau+\int_{0}^{t}\gamma_{i2}(t-\tau )[e_{1}e_{6}+e_{2}e_{5}]d\tau\\ &\quad+\int_{0}^{t}\gamma_{i3}(t-\tau)[e_{2}e_{7}+e_{3}e_{6}]d \tau,\quad i=1,2,3,4,\end{split}