## 1 Introduction  
The squeeze film damper (SFD), serving as a critical vibration-suppression component in high-speed turbomachinery, enhances system stability and reliability by providing supplementary damping. Due to its unique vibration-suppression mechanism and exceptional performance characteristics, SFD has become an indispensable core component in modern turbomachinery, including aeroengines, gas turbines, and compressors. However, under improper parameter design or adverse operating conditions, the inherent nonlinear characteristics of SFD may trigger system instability, significantly affecting operational performance [1, 2, 3]. For nonlinear rotor stability analysis, Floquet theory provides a rigorous mathematical foundation for examining the stability of periodic orbits, enabling quantitative stability assessment and prediction of potential bifurcation behavior through the calculation of Floquet multipliers. Regarding computational methods based on Floquet theory, [PERSON] [4] developed an improved Runge-Kutta algorithm for computing the Floquet monodromy matrix based on [PERSON]'s method [5]. This method significantly enhanced computational efficiency while maintaining accuracy and has been successfully applied in helicopter rotor blade stability analysis. [PERSON] et al. [PERSON] et al. (2016) systematically compared five stability computation methods based on Floquet theory for periodic solutions, including the shooting method, [PERSON]'s method, explicit Runge-Kutta integration, implicit Newmark integration, and [PERSON]'s method. Through analyses of both Jeffcott rotor and multi-degree-of-freedom finite element rotor models, the results indicated that the Newmark method demonstrated the most balanced performance in terms of computational efficiency and precision. Extensive research has been conducted on the nonlinear behavior and stability analysis of rotor systems with SFDs through the application of Floquet theory. [PERSON] et al. [PERSON] et al. (2016) revealed that the jump phenomenon in rigid rotor systems with eccentric SFD essentially represents a saddle-node bifurcation, demonstrating that large static eccentricity and mass unbalance can lead to subharmonic and quasi-periodic responses. [PERSON] et al. [PERSON] et al. (2016) developed an improved harmonic balance method combined with [PERSON]'s method for computing Floquet multipliers, numerically and experimentally validating nonlinear phenomena, including subharmonic resonance, superharmonic resonance, and quasi-periodic motions in eccentric SFD rotor systems. [PERSON] (2016) employed numerical integration to investigate the stability effects of unbalance, bearing stiffness, and journal mass on single-disk flexible rotor systems with SFD, identifying period-four and quasi-periodic motions under specific parameter combinations. [PERSON] et al. [PERSON] et al. (2016) innovatively combined the harmonic balance method with Newton-Raphson iteration, proposing a novel transfer matrix method and deriving equations for Jeffcott rotor systems with SFD, analyzing systematic stability through Floquet theory. [PERSON] (2016) examined support stiffness effects on the stability of Jeffcott rotor systems with elastically supported SFD, revealing that system responses alternate between periodic and quasi-periodic motions within specific speed ranges. In contrast, excessive support stiffness induces 1/4 and 1/2 subharmonic components. Researchers have conducted investigations into the stability behavior of SFDs under dual-frequency excitation and in coupling with other nonlinear elements. [PERSON] et al. [PERSON] et al. (2016) investigated the jump mechanisms of dual-rotor systems with SFD under dual-frequency excitation using the harmonic balance method combined with Runge-Kutta numerical integration, discovering coexisting soft and hard characteristics near the third-order resonance peak. The system transitions from periodic to quasi-periodic solutions through secondary Hopf bifurcation, with increased excitation frequency ratios enhancing system stability.