## 2 Mathematical Model
\begin{array}{l}\mathbf{d}\;=\;-(\frac{A}{\Delta\Omega^{3}}-\frac{2\Pi}{\Delta t^{2}}) \mathbf{\Phi_{4}}+(\frac{2A}{\Delta t^{2}}-\frac{5\Pi}{\Delta t^{3}})\mathbf{\Phi}_{3}+ \big{(}\frac{3\Pi}{\Delta t^{2}}-\frac{A}{\Delta t}\big{)}\mathbf{\Phi}_{2},\\ \overline{\mathbf{d}}\;=\;(\frac{A}{\Delta t^{3}}-\frac{2\Pi}{\Delta t^{2}})\mathbf{ \Phi_{4}}+(\frac{3\Pi}{\Delta t^{3}}-\frac{A}{\Delta t^{2}})\mathbf{\Phi_{3}},\\ \mathbf{H_{1}}\;=\;(\frac{\mathbf{\Omega}}{\Delta t^{3}}-\frac{3\Pi}{\Delta t^{2}})\mathbf{ \Phi_{3}}+\mathbf{\Phi_{0}}+(\mathbf{A}-\frac{1}{\Delta t})\mathbf{\Phi_{1}}+(\frac{3\Pi}{ \Delta t}-\frac{3\Pi}{\Delta t^{2}})\mathbf{\Phi_{2}}-(\frac{2\Pi}{\Delta t^{4}}+ \frac{A}{\Delta t^{3}})\mathbf{\Phi_{4}},\\ \mathbf{H_{2}}\;=\;\frac{\mathbf{\Omega}}{\Delta t}+\frac{\mathbf{\Phi_{0}}}{\Delta t}+(- \frac{2A}{\Delta t^{2}}-\frac{3\Pi}{\Delta t^{3}})\mathbf{\Phi_{3}}+(\frac{2\Pi}{ \Delta t}+\frac{A}{\Delta t^{3}})\mathbf{\Phi_{4}},\\ \mathbf{e}_{j,k,l}\;=\;(-1+\frac{3\mu_{j}}{2\Delta t^{3}}-\frac{\mu_{j}^{2}}{\Delta t ^{2}})\mathbf{\Phi_{0}}+\frac{\mathbf{\Phi_{3}}}{2\Delta t^{3}}+(-\frac{2\Pi}{\Delta t ^{2}}+\frac{\mu_{j}\Delta t}{\Delta t^{2}})\mathbf{\Phi_{2}}+(\frac{5}{2\Delta t} -\frac{5\mu_{j}\Delta t}{2\Delta t}+\frac{\mu_{j}^{2}\Delta t}{2\Delta t})\mathbf{ \Phi_{1}},\\ \overline{e}_{j,k,l}\;=\;-\frac{\mathbf{\[PERSON]_{3}}}{2\Delta t^{3}}+(\frac{3}{2 \Delta t^{2}}-\frac{\mu_{j,k,l}}{\Delta t^{2}})\mathbf{\[PERSON]_{2}}+(-\frac{1}{ \Delta t}+\frac{3\mu_{j,l}}{2\Delta t}-\frac{\mu_{j,l}^{2}}{2\Delta t})\mathbf{ \[PERSON]_{1}},\\ \mathbf{f}_{j,k,l}\;=\;-\frac{\mathbf{\[PERSON]_{3}}}{\Delta t^{3}}+(-2w_{j,k,l}+w_{j,k,l}^ {2})\mathbf{\[PERSON]_{0}}+(\frac{3}{\Delta t^{2}}-\frac{2\mu_{j,k,l}}{\Delta t^{2}}) \mathbf{\[PERSON]_{2}}+(-\frac{2}{\Delta t}+\frac{4w_{j,k,l}}{\Delta t}-\frac{w_{j,l}^ {2}}{\Delta t^{2}})\mathbf{\[PERSON]_{1}},\\ \widetilde{f}_{j,k,l}\;=\;\frac{\mathbf{\[PERSON]_{3}}}{\Delta t^{3}}+(-\frac{2\mu_{j, k,l}}{\Delta t^{2}})\mathbf{\[PERSON]_{2}}+(-\frac{2\mu_{j,l}}{\Delta t}+\frac{w_{j,l}^ {2}}{\Delta t})\mathbf{\[PERSON]_{1}},\\ \mathbf{g}_{j,k,l}\;=\;\frac{\mathbf{\[PERSON]_{3}}}{2\Delta t^{3}}+(\frac{\mu_{j,k}}{2 \Delta t^{2}}-\frac{\mu_{j,k,l}}{\Delta t^{2}})\mathbf{\Phi_{0}}+(-\frac{1}{ \Delta t^{2}}+\frac{w_{j,k,l}^{2}}{\Delta t^{2}})\mathbf{\[PERSON]_{2}}+(\frac{1}{2 \Delta t}-\frac{3w_{j,k,l}}{2\Delta t}+\frac{w_{j,l}^{2}}{2\Delta t})\mathbf{\Phi _{1}},\\ \overline{\mathbf{g}}_{j,k,l}\;=\;-\frac{\mathbf{\Phi_{3}}}{2\Delta t^{3}}+(\frac{1}{ 2\Delta t^{2}}-\frac{w_{j,l}}{\Delta t^{2}})\mathbf{\Phi_{2}}+(\frac{w_{j,k,l}}{2 \Delta t}-\frac{w_{j,k,l}^{2}}{2\Delta t})\mathbf{\Phi_{1}}.\end{array}  \tag{13}\]  
Now, if defining state vector \(\mathbf{V}_{i}\;=\;\left[\mathbf{x}_{i},\mathbf{x}_{j-1}\cdots,\mathbf{x}_{i-MX+1},\mathbf{x}_{i-MX }\right]^{T}\) with \(MX=\max(m_{j,k,i})\), Equation (8) can be recast into a discrete map form [7; 8; 9] as  
\[\mathbf{V}_{i+1}\;=\;\mathbf{Z}_{i}\mathbf{V}_{i} \tag{14}\]  
where \(\mathbf{Z}_{i}\) is the coefficient matrix in the form of  
\[\mathbf{Z}_{i}\;=\;\left[ \begin{array}{ccccc}\mathbf{Q}_{i}^{-1}\mathbf{H}_{i}&0&\cdots&0 &0&0\\ \mathbf{I}&0&\cdots&0&0&0\\ 0&\mathbf{I}&\cdots&0&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&\cdots&\mathbf{I}&0&0\\ 0&0&\cdots&0&\mathbf{I}&0\\ \end{array} \right]+\sum\limits_{j=1}^{N}\sum\limits_{k=1}^{N}\left[ \begin{array}{ cccccc}0&\cdots&\mathbf{Q}_{i}^{-1}\mathbf{H}_{i-m_{j,k,j}+2}&\mathbf{Q}_{i}^{-1}\mathbf{H}_{i-m_{j,k,j}+1}& \mathbf{Q}_{i}^{-1}\mathbf{H}_{i-m_{j,k,j}}&\cdots&0\\ 0&\cdots&0&0&0&\cdots&0\\ 0&\cdots&0&0&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&\cdots&0&0&0&\cdots&0\\ 0&\cdots&0&0&0&\cdots&0\\ \end{array}