## 1 Introduction  
Vibrations occur in many engineering applications. In most cases, they are undesirable and harmful, as they shorten the service life of equipment and even cause serious accidents [1]. Recently, with the rapid development of sophisticated industry, such as high-speed trains [2], precision optical instruments [3] and stealth submarines [4], the requirements on the vibration environment have become increasingly strict. Considering the contradiction between the load-bearing capacity and the low-frequency vibration isolation of traditional linear vibration isolators, nonlinear theory is widely adopted for better isolation performance [5]. Nonlinear vibration isolation systems can be implemented without additional energy, and constructed in a variety of interesting ways. Air springs [6], polycal wire ropes [7], bio-inspired X-shaped structures [8], and fluidic origami structures [9] are all effective designs to achieve the expected vibration isolation. The use of a negative stiffness corrector (NSC) to develop nonlinear isolators on the basis of linear isolators is also an attractive issue in the vibration control field. In this type of nonlinear vibration isolator, the static stiffness is high, but the dynamic stiffness drops to extremely low and can even realize the quasi-zero stiffness (QZS). This means that introducing an NSC will reduce the natural frequency without increasing the static deflection of the original system, which is beneficial to vibration isolation. The early NSC, whose dynamic model was established and fully analyzed in [10; 11], consists of a pair of inclined linear springs. Buckled beams are another choice for negative stiffness structure due to their elastic property [12; 13]. Nonlinear magnet metamaterials and honeycomb configurations are also incorporated in negative stiffness structures to reduce friction and deformation during vibration isolation [14] or to isolate shocks [15]. As the theory of negative stiffness becomes mature, the use of NSCs to suppress vibration in many practical applications have been discussed. [PERSON] et al. [16] installed the vibration isolator with a negative stiffness element in a passenger vehicle. In their research, five different isolators were compared under different seated passengers and excitations from rough roads. [PERSON] et al. [17] introduced a negative stiffness adjusting mechanism into the suspension system of trucks with an all-floating cab. Their simulation results presented that the vibration of the cab with the nonlinear suspension system is significantly attenuated, and the suspension travel space is utilized due to the controllable deflection. [PERSON] and [PERSON] [18] conducted their study on a double-layer QZS vibration isolation floating raft system which expands the isolation region to a very low-frequency: [PERSON] et al. [19] used permanent magnets as a negative stiffness corrector to construct the QZS vibration isolator, aiming to weaken the vibrations of the infant in an incubator during neonatal transport. For the flexible structures, such as stay cables, a negative stiffness damper was proposed to mitigate the damage caused by vibrations. [PERSON] and [PERSON] [20] investigated the matching of the negative stiffness and viscous damping in that passive equipment and revealed its vibration control mechanism from the perspective of energy. Clearly, it has been well proven that adding the NSC to a vibration isolator can improve the isolation performance. In literature, however, the vibration isolator applying an NSC often has the nonlinear stiffness with a complex analytical expression [10; 11; 14; 16; 17; 18; 19; 21; 22; 23; 24; 25; 26; 27] which needs to be approximated by the Taylor series expansion before solving its dynamic characteristics, including frequency response function, bounded response area and peak amplitude. In fact, the strong nonlinearity makes the vibration isolator with NSC sensitive to parameters and prone to instability.