## 1 Introduction  
[PERSON] [1] previously recognized the phenomenon of parametric resonance when he studied the response frequency of surface waves in a fluid-filled cylinder with vertical excitations. Researchers have subsequently carried out extensive investigations of parametric resonances for various nonlinear vibration systems in the field of neutrino oscillations [2], quantum fields [3], ship dynamics [4], Brownian dynamics of water droplets [5], bubbles oscillating in water [6], micro-electro mechanical systems [7, 8], and rotor dynamics [9, 10]. Therefore, the topic of parametric resonance is a basic scientific problem in the field of nonlinear dynamics. Rolling bearings are widely used in rotating machineries due to their advantages of lower friction compared to sliding bearings and ease with which they can be selected from a manufacturer's catalogue [11]. There are two major effects of a healthy rolling bearing on machine vibration [12, 13]:spring characteristics and the excitation force on the supported structure. In the dynamic design of the bearing-rotor system, the excitation characteristics of rolling bearings should be given adequate consideration. Varying compliance (VC) excitation is induced when rolling elements move along the bearing raceway with an external load, which results in time-varying bearing stiffness in the rotor system. Thus, VC excitation is an inevitable case of parametric excitation [9; 14]. [PERSON] [9] found that VC excitation can result in periodic and irregular non-periodic vibrations in cylindrical roller bearings using digital simulation and experiments. [PERSON] et al. [14] identified that VC vibration may exhibit subharmonic, quasi-periodic, chaotic, and beat motions at critical speeds, considering a two-degree-of-freedom ball bearing model with Hertzian contact nonlinearity. [PERSON] and [PERSON] [15; 16] studied the routes to chaos in ball bearings and noted the chaos due to period doubling in the resonant speed range. [PERSON] [17] found that the VC vibration amplitude of rolling bearings is directly related to the bearing deflection in two specific boundary positions of the rolling elements, and a simplified rolling bearing model for description of VC vibrations was presented. Regarding VC hysteresis characteristics, [PERSON] et al. [18] found a hysteretic amplitude-frequency response in ball bearing VC vibration using an early shooting method. [PERSON] et al. [19] investigated the hysteresis behaviors using static bifurcation analysis and provided closed-form expressions of VC periodic solutions using the generalized averaging method. Based on Hertzian contact resonance [20; 21], [PERSON] et al. [22; 23; 24; 25] investigated the mechanism, influence, and suppression of VC resonant hysteresis in ball bearings. Theoretical and experimental results showed that the primary contact resonance exhibits soft spring behavior in the direction exerting constant force, and the coexistence of soft and hard spring characteristics in the direction with zero external force. In addition, [PERSON] et al. [26] noted a case of complex coupling VC resonance caused by the superposition of VC parametric primary resonance and 1:2 internal resonance. The above research is mainly focused on the primary VC resonant responses. However, it should be noted that a small parametric excitation may produce remarkable responses, even if the excitation frequency significantly differs from the linear natural frequencies of the system [27]. Other parametric resonance types may also exist in a parametric system, such as superharmonic, subharmonic, and combined cases. Rapid, flexible, and automated development requires higher stability and safety characteristics in rotating machineries, which promotes the study of super- and subharmonic resonances, and their combinations, in rolling bearing-rotor systems. [PERSON] et al. [28; 29], [PERSON] et al. [30; 31], and [PERSON] et al.