## 3 Nonlinear Motions of the Brush Seal Rotor System
### Bifurcation Characteristics  
We apply a discrete mapping method [19; 20] to the brush seal rotor system for the bifurcations and nonlinear motions. The discrete mapping method employs implicit mapping structures for nonlinear motion solutions, which are so called semi-analytical solutions. The Jacobian matrix for the bifurcation analysis can be obtained as follows. \[ \begin{array}{ll}\mathbf{DP}&=\left[\frac{\partial\mathbf{x}_{N}}{\partial \mathbf{x}_{0}}\right]_{(\mathbf{x}_{0}^{*},\mathbf{x}_{0}^{*},\cdots,\mathbf{ x}_{N}^{*})}=DP_{N}\cdot DP_{N-1}\cdot\ldots\cdot DP_{2}\cdot DP_{1}\\ &=\prod\limits_{k=N}^{1}DP_{k}\end{array}  \tag{10}\]  
where  
\(x_{1,k}\) and \(x_{2,k}(k=0,1,2,\cdots,+\infty)\) are the discrete node points of the displacement and velocity in the \(x\)-direction. And \(y_{1,k}\) and \(y_{2,k}(k=0,1,2,\cdots,+\infty)\) are the node points of the displacement and velocity in the \(y\)-direction. The detailed components of the above Jacobian matrix have the following expressions. The initial derivation is put in the Appendix A. \[ \begin{array}{l}\frac{\partial x_{1k}}{\partial x_{1j\lambda-1}}=1+\frac{1}{2}h \frac{\partial x_{2k}}{\partial x_{1j\lambda-1}},\;\frac{\partial x_{2k}}{ \partial x_{1j\lambda-1}}=\frac{(\Delta_{11}+\Delta_{13})\Delta_{41}+\frac{1} {2}h(\Delta_{22}+\Delta_{23})\Delta_{12}}{\Delta_{31}\Delta_{41}-\frac{1}{2}h^ {2}\Delta_{12}\Delta_{22}},\\ \frac{\partial x_{1 jk}}{\partial x_{2k-1}}=\frac{1}{2}h(\frac{\partial x_{2k}} {\partial x_{2k-1}}+1),\;\frac{\partial x_{2k}}{\partial x_{2k-1}}=\frac{( \frac{1}{2}-\frac{1}{2}+\frac{1}{2}h\Delta_{11})\Delta_{41}+\frac{1}{2}h^{2} \Delta_{12}\Delta_{22}}{\Delta_{31}\Delta_{41}-\frac{1}{2}h^{2}\Delta_{12} \Delta_{22}},\\ \frac{\partial x_{1 jk}}{\partial y_{1k-1}}=\frac{1}{2}h\frac{\partial x_{2k}} {\partial y_{1k-1}},\;\frac{\partial x_{2k}}{\partial y_{1k-1}}=\frac{(\Delta _{12}+\Delta_{41})\Delta_{41}+\frac{1}{2}h\Delta_{12}(\Delta_{21}+\Delta_{22} )}{\Delta_{31}\Delta_{41}-\frac{1}{2}h^{2}\Delta_{22}},\\ \frac{\partial x_{1 jk}}{\partial y_{2k-1}}=\frac{1}{2}h\frac{\partial x_{2k}} {\partial y_{2k-1}},\;\frac{\partial x_{2k}}{\partial y_{2k-1}}=\frac{\frac{1} {2}h\Delta_{21}\Delta_{41}+\frac{1}{2}h\Delta_{12}(\frac{1}{2}-\frac{1}{2}h^ {2}\Delta_{12})}{\Delta_{31}\Delta_{41}-\frac{1}{2}h^{2}\Delta_{12}\Delta_{22 }},\;a\\ \frac{\partial y_{1 jk}}{\partial x_{1 jk-1}}=\frac{1}{2}h\frac{\partial y_{2k}} {\partial x_{1 jk-1}},\;\frac{\partial y_{2k}}{\partial x_{1k-1}}=\frac{( \Delta_{22}+\Delta_{22})\Delta_{31}+\frac{1}{2}h(\Delta_{11}+\Delta_{13}) \Delta_{22}}{\Delta_{31}\Delta_{41}-\frac{1}{2}h^{2}\Delta_{22}},\\ \frac{\partial y_{1 jk}}{\partial x_{2k-1}}=\frac{1}{2}h\frac{\partial y_{2k}} {\partial x_{2k-1}},\;\frac{\partial y_{2k}}{\partial x_{2k-1}}=\frac{\frac{1 }{2}h\Delta_{22}\Delta_{31}+\frac{1}{2}h\Delta_{22}(\frac{1}{2}-\frac{1}{2}h^ {2}\Delta_{11})}{\Delta_{31}\Delta_{41}-\frac{1}{2}h^{2}\Delta_{22}},\\ \frac{\partial y_{1 jk}}{\partial y_{1 jk-1}}=1+\frac{1}{2}h\frac{\partial y_{2k} }{\partial y_{1 jk-1}},\;\frac{\partial y_{2k}}{\partial y_{1k-1}}=\frac{( \Delta_{21}+\Delta_{22})\Delta_{31}+\frac{1}{2}h\Delta_{22}(\Delta_{12}+ \Delta_{41})}{\Delta_{31}\Delta_{41}-\frac{1}{2}h^{2}\Delta_{12}\Delta_{22}}, \\ \frac{\partial y_{2 jk}}{\partial y_{2k-1}}=\frac{1}{2}h(\frac{\partial y_{2k}} {\partial y_{2k-1}}+1),\;\frac{\partial y_{2k}}{\partial y_{2k-1}}=\frac{( \frac{1}{2}-\frac{1}{2}+\frac{1}{2}h\Delta_{21})\Delta_{31}+\frac{1}{2}h^{2} \Delta_{12}\Delta_{22}}{\Delta_{31}\Delta_{41}-\frac{1}{2}h^{2}\Delta_{12} \Delta_{22}}.\end{array}  \tag{12}\]  
with  
\[