## 3 Nonlinear Motions of the Brush Seal Rotor System
### Bifurcation Diagrams
The eigenvalue analysis within \(\Omega\in(5.6,10.1)\) is plotted in Figure 8a,c,e. Because the nonlinear motions within \(\Omega\in(0.0,5.6)\) are all stable, the eigenvalues within this speed range are all with magnitudes less than one, so this part is skipped. During the increasing speed process, a period-doubling bifurcation occurs at \(\Omega=9.23\) with one eigenvalue greater than one, one eigenvalue crossing the negative-one line, and the others smaller than one so that unstable period-2 motions may occur with small perturbation. Such unstable period-2 motions come from unstable period-1 motions. The nonlinear rotor usually accompanies dangerous motions with large displacement. A stable period-doubling bifurcation happens at \(\Omega=7.85\), where period-4 motions will happen. The eigenvalue analysis and bifurcations within \(\Omega\in(10.0,15.0)\) are presented in Figure 8b,d,f. Two period-doubling bifurcations are discovered at \(\Omega=11.47\) and \(\Omega=12.32\) where period-2 and 4 motions will be produced during running. The real, imaginary, and magnitude parts of the eigenvalues provide a quantitative understanding of the bifurcations in the brush seal rotor system. Figure 8: Bifurcation analysis by eigenvalues. I. \(\Omega\in(5.6,10.1)\): (**a**) real parts, \(\text{Re}(\lambda_{i})\): (**c**) imaginary parts, \(\text{Im}(\lambda_{i})\): (**e**) magnitudes, \(\text{Mag}(\lambda_{i})\); II. \(\Omega\in(10.0,15.0)\): (**b**) real parts, \(\text{Re}(\lambda_{i})\): (**d**) imaginary parts, \(\text{Im}(\lambda_{i})\): (**f**) magnitudes, \(\text{Mag}(\lambda_{i})\). In Figures 9 and 10, the semi-analytical Poincare maps of period-2 motions are presented to illustrate the bifurcations and nonlinear motions. The solid lines mean stable motions and the dashed lines mean unstable motions. When the speed is increasing, the period-1 motion is stable. A saddle node bifurcation happens at \(\Omega=5.917\) to generate chatting during the speed-increasing process. A Neimark bifurcation happens at \(\Omega=6.53\) and an unstable quasi-period-1 motion happens. The period-doubling bifurcation at \(\Omega=9.23\) causes a period-2 motion, and half-frequency components in the nonlinear motions emerge. Another period-doubling bifurcation at \(\Omega=7.85\) causes a period-4 motion, and quarter-frequency components in the nonlinear motions emerge. Moreover, quasi-periodic period-2 motions are produced at Neimark bifurcations on \(\Omega=10.37\) and \(\Omega=11.08\). Nonlinear chattering and jumping motions are produced at saddle node bifurcations at \(\Omega=10.197\) and \(\Omega=14.307\). Period-2 and period-4 motions are produced at period-doubling bifurcations on \(\Omega=11.47\) and \(\Omega=12.32\) in the brush seal rotor system. The nonlinear motions behave stably and unstably as well as at complex bifurcations. Attention should be paid to the practical rotor design to put the working speed range smaller than these stability shifts. Care must be taken in operation to avoid unstable and dangerous nonlinear motions. Figure 9: Semi-analytical Poincaré map of the nonlinear period-2 motions in the rotor system within \(\Omega\in(5.6,10.1)\): (a) Poincare displacement, \(x_{1,k}\); (b) Poincare velocity, \(x_{2,k}\); (c) Poincare displacement, \(y_{1,k}\); (d) Poincare velocity, \(y_{2,k}\); \((k=0,1,2,\cdots,)\).