## 3 The Synchronization Condition for Exact LS between Systems (2) and (3)
 \tag{15}\]  
where \(\gamma_{ij}(t)=L^{-1}[\frac{h_{ij(s)}}{D(s)}]\), \(i=1,2,3,4\) and \(j=0,1,2,3\). If the denominator \(D(s)\) in Equation (14) can be decomposed into \(D(s)=Q_{1}(s)Q_{2}(s)\), and \(Q_{1,2}\) have no common factors, then \(h_{ij}(s)/D(s)\), \(i\in\{1,2,3,4\}\), \(j\in\{0,1,2,3\}\) can be decomposed into  
\[\frac{h_{ij}(s)}{D(s)}=\frac{\bar{h}_{ij}(s)}{Q_{1}(s)}+\frac{\hat{h}_{ij}(s)} {Q_{2}(s)},\quad i\in\{1,2,3,4\},\,j\in\{0,1,2,3\},\]  
where \(\bar{h}_{ij}(s)\) and \(\hat{h}_{ij}(s)\) are polynomials of \(s\). Moreover, the highest powers are less than those of \(Q_{1}(s)\) and \(Q_{2}(s)\). One can continue the process until only the following two classes of functions retain  
\[\frac{A_{1}(s)}{(s-\eta_{0})^{m}},\quad\frac{A_{2}(s)}{(s^{2}+\eta_{1}s+\eta_{ 2})^{n}},\]  
in which \(m\in\{1,2,3,4\}\), \(n\in\{1,2\}\), \(\eta_{0,1,2}\in R\), \(\eta_{1}^{2}-4\eta_{2}<0\) and the highest powers of polynomials \(A_{1,2}\) are less than \(m\) and \(n\), respectively. Consider the inverse Laplace transform in the following three cases:  
* \(\frac{A_{1}(s)}{(s-\eta_{0})^{m}}=\frac{B_{1}}{s-\eta_{0}}+\frac{B_{2}}{(s-\eta _{0})^{2}}+\cdots+\frac{B_{m}}{(s-\eta_{0})^{m}},\) where \(B_{(m-s)}=\left.\frac{1}{1!}\frac{d!}{ds! }\right[\frac{A_{1}(s)}{(s-\eta_{0})^{ m}}(s-\eta_{0})^{m}]\bigg{|}_{s=\eta_{0}},\quad i=1,2,\cdots,m-1,\) \[B_{m}=\left.\frac{A_{1}(s)}{(s-\eta_{0})^{m}}(s-\eta_{0})^{m}\right|\bigg{|}_{s= \eta_{0}},\] \[L^{-1}[\frac{A_{1}(s)}{(s-\eta_{0})^{m}}]=(B_{1}+B_{2}t+B_{3}t^{2 }+\cdots+B_{m}t^{m-1})e^{\eta_{0}t}\]\[\bullet \quad\frac{A_{2}(s)}{s^{2}+\eta_{1}s+\eta_{2}}=\frac{A_{2}(s)}{(s- \omega_{1}-\omega_{2})(s-\omega_{1}+\omega_{2})}=\frac{B_{1}}{s-\omega_{1}- \omega_{2}}+\frac{B_{2}}{s-\omega_{1}+\omega_{2}},\] \[\quad\text{where}\quad B_{1,2}=\frac{A_{2}(s)}{2s+\eta_{1}}\Big{|}_ {s=\omega_{1}\pm i\omega_{2}},\] \[L^{-1}[\frac{A_{2}(s)}{s^{2}+\eta_{1}s+\eta_{2}}]=B_{1}e^{( \omega_{1}+\omega_{2})t}+B_{2}e^{(\omega_{1}-i\omega_{2})t}\] \[\bullet \quad\frac{A_{2}(s)}{(s^{2}+\eta_{1}s+\eta_{2})^{2}}=\frac{B_{1} }{(s-\omega_{1}-\omega_{2})^{2}}+\frac{B_{2}}{s-\omega_{1}-\omega_{2}}+\frac{ B_{3}}{(s-\omega_{1}+\omega_{2})^{2}}+\frac{B_{4}}{s-\omega_{1}+i\omega_{2}},\] \[\quad\text{where}\quad B_{1,3}=\frac{A_{2}(s)}{s^{2}+\eta_{1}s+ \eta_{2}}(s-\omega_{1}\pm j\omega_{2})^{2}\Big{|}_{s=\omega_{1}\pm j\omega_{2 }},\] \[B_{2\mathcal{A}}=\frac{d}{ds}[\frac{A_{2}(s)}{s^{2}+\eta_{1}s+ \eta_{2}}(s-\omega_{1}\pm j\omega_{2})^{2}]\Big{|}_{s=\omega_{1}\pm j\omega_{ 2}},\] \[\frac{A_{1}(s)}{(s-\eta_{0})^{m}},\quad\frac{A_{2}(s)}{(s^{2}+\eta_{1}s+\eta_{ 2})^{n}},\]0  
From Equation (15), the necessary conditions for \(||e_{i}||\to 0\), \(i=1,2,3,4\) are \(\gamma_{ij}(t)\to 0\), \(i\in\{1,2,3,4\}\) and \(j\in\{0,1,2,3\}\). According to the above analysis for the inverse Laplace transform, all eigenvalues of matrix (11) should have negative real parts. Under such conditions, when \(t\to 0\) system (15) becomes  
\[ \begin{split} e_{i}=&-\int_{0}^{t}\gamma_{i1}(t- \tau)[e_{1}e_{7}+e_{3}e_{5}]d\tau+\int_{0}^{t}\gamma_{i2}(t-\tau)[e_{1}e_{6}+ e_{2}e_{5}]d\tau\\ &+\int_{0}^{t}\gamma_{i3}(t-\tau)[e_{2}e_{7}+e_{3}e_{6}]d\tau, \quad i=1,2,3,4.\end{split}  \tag{16}\]  
Actually, these are a set of Volterra integral equations in system (16) that can be solved based on the successive approximation method [30]. For the sake of clarity, we briefly introduce the successive approximation method below. Consider the integral equation of the following form  
\[y(t)=\Phi(t)+\int_{0}^{t}g(t-\tau)h(\tau,y(\tau))d\tau, \tag{17}\]  
where \(g\) is an \(n\times n\) matrix; \(\Phi(t)\) and \(h(t,y(t))\) are vectors with \(n\) components. If the following conditions are satisfied, then Equation (17) has a unique continuous solution:  
* \(|y|<\infty\);
* \(\Phi\) and \(h\) are continuous for \(0<t<t_{0}\), in which \(0<t_{0}<+\infty\);
* \(|g|\in L[0,\epsilon]\) holds for any \(0<\epsilon<t_{0}\);
* For any \(\xi>0\), if \(|y_{1}-y_{2}|<\xi\) there must exist a constant \(\epsilon>0\) such that \(|h(t,y_{1})-h(t,y_{2})|<\epsilon\).