## 4 Conclusions  
In this research, the elasto-dynamic theory was utilized to model brush seal forces. A nonlinear brush seal rotor system was constructed by combination of a flexible rotor system with nonlinear brush seal forces. The nonlinear motions were obtained by a semi-analytical approach as well as stability and bifurcations. The system chattered nonlinearly at specific speeds near saddle node bifurcations. Parameter maps of bifurcations and stability domains of the brush seal rotor system were also achieved. Complicated vibrations and bifurcations were successfully obtained by eigenvalues and exhibited in semi-analytical Poincare maps. Period-doubling bifurcations were obtained to lead the route from period-1 motion to period-2 motion. Numerical comparisons were performed to verify the theoretical results. The period-doubling bifurcations caused motion switch and subharmonic-1/2 vibrations in the brush seal rotor system. The saddle node bifurcations on the parameter map make it possible for the system to prevent nonlinear motion jumping and reduce the harmful motions if optimal parameters are considered. Nonlinearity is an avoidable effect in rotor system design and production. With the increase in the complexity of the practical rotor system, this research provides a method to solve nonlinear resonance, stability, and bifurcations, which will be helpful in future designs to avoid dangerous vibrations. Methodology, Y.X. ; Validation, Y.Z. ([PERSON]); Investigation, Y.Z. ([PERSON]); Writing--review & editing, Z.X. ; Supervision, Y.X. ; Funding acquisition, M.W. and D.L. All authors have read and agreed to the published version of the manuscript. This work is supported by the Jilin Provincial Natural Science Foundation (Grant No. 20230101208 JC), the Key R&D and Transformation Plan Project of Qinghai Province (No. 2023-QY-215), National Nature Science Foundation of China (Grant No. 12102319), and the Fundamental Research Funds for the Central Universities (Grant No. xzy012021004). The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. The authors declare they have no conflict of interest. Figure 11: Parameter maps of bifurcations in the brush seal rotor system: (**a**) parameter map of period-doubling bifurcations; (**b**) parameter map of saddle node bifurcations. [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Numerical study on heat transfer mechanism of brush seal based on three-dimensional solid modeling. _J. Aerosp. Power_**2019**, _34_, 1633-1643. * _[PERSON] et al. (2019)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Theory and experiment on the leakage characteristics of brush seals based on fluid-structure interaction. _J. Aerosp. Power_**2019**, _34_, 1519-1529. * _[PERSON] et al. (2015)_ [PERSON]; [PERSON]; [PERSON] Moving force-induced motion of a rotating beam with elastic boundary conditions. _Int. J. Struct. Stab. Dyn._**2015**, _15_, 1540035. [CrossRef]
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