**1. Introduction**

In the metallurgical industry, the rolling mill represents a core piece of equipment. However, it has been found that there are always some parameter changes or disturbance factors during the rolling process, which may induce vibration in the rolling mill [1,2]. This may even cause the load roll system to produce vertical vibration with large amplitude, leading to production accidents such as steel-heaping, strip breakage, component damage and so on [3–5]. The hydraulic automatic gauge control (HAGC) system is considered the core control system of a rolling mill. The stability of the HAGC system is crucial to ensuring a rolling process with high speed, high precision and high reliability.

For a long time, many scholars and research institutions have explored the rolling mill's vibration characteristics, and rich results have been demonstrated. Zhong et al. [6] explored the mechanism of interfacial coupling and electromechanical coupling in the vibration of rolling mills. It was found that the various vibration phenomena of a high-speed rolling mill were related to various interactions of the rolling process. Wang et al. [7] conducted a series of studies on the control of plate thickness and plate shape, which laid a theoretical foundation for improving the control accuracy of the HAGC system. Chen et al. [8] studied the nonlinear parametric vibration of a four-roll cold strip mill and explored the influence of the deformation resistance of rolled piece on parametric resonance. The effects of rolling speed and tension on the nonlinear vibration of the rolling mill were investigated by Sun et al. [9], and the influence of external excitation frequency on system stability was analyzed. Liu et al. [10] researched the vertical, nonlinear parametric vibration characteristics of the mill roll system, and discussed the influence of parameters such as damping, stiffness and external excitation on system stability. In consideration of the change of parameters, Liu et al. [11] analyzed the partial bifurcation phenomenon of the load roll system, finding that both internal resonance and main resonance have an amplitude jump phenomenon. Yan et al. [12] studied the coupling vibration mechanism and vibration suppression method of the rolling mill. The characteristics and laws of vibration of the hot rolling mill were studied by the analysis of the measured signals, both in the time and frequency domain [13,14]. Yang et al. [15] researched the nonlinear modeling and stability of the vertical vibration of a cold rolling mill. The singular value theory was used to analyze the stability of the system. Moreover, the effects of parameters such as damping and stiffness on the vibration characteristics of the system were researched. In terms of periodic excitation, Bi et al. [16,17] studied the bifurcation mechanism of oscillations of the dynamic system. Wang and Qian et al. [18,19] investigated the effects of important components such as pump [20–23] and valve [24,25] on the vibration characteristics. Bai et al. [26–28] researched the vibration of the pump under varied conditions. Zhang et al. [29–31] studied the influence of excitation forces on the vibration of the pump and the measure of noise reduction. Niziol et al. [32] investigated the effect of rolling speed on the vibration of a mill roll system, and analyzed the steady state domain of the vibration of the roller system under different rolling speeds. Heidari et al. [33,34] studied the influence of friction damping and lubrication state on the flutter of cold strip mill, and analyzed the effects of some main parameters of lubricant on the critical vibration velocity. In order to study the self-excited vibration and nonlinear paramagnetic vibration of the rolling process, a nonlinear mathematical model for the vibration of the rolling mill was established by Drzymala et al. [35].

The above research results have provided a theoretical guidance for the vibration mechanism analysis and vibration suppression of the rolling mill system. Nevertheless, based on the existing research results, some problems have been discovered. In most of the research results, the influence of the HAGC system is often overlooked when the vertical vibration of the rolling mill is analyzed [36,37]. In essence, the HAGC system is a nonlinear closed-loop system, which has many factors affecting its stability. If it is unstable, it will certainly have an influence on the vibration characteristics of the load roll system [38,39]. In addition, with the development of nonlinear science, it is of important theoretical significance to reveal the essence and mechanism of the system dynamic process by applying the nonlinear dynamics theory, which simultaneously presents broad application prospects [40–42]. This area is worthy of further study when the theory and method of nonlinear dynamics are utilized to explore the vibration mechanism of the HAGC system, and then to obtain the influence law and incentive of nonlinear factors on the vibration characteristics of the system.

In this paper, based on the theory and method of nonlinear dynamics, the influence of some nonlinear parameters of the HAGC system on load vertical vibration will be researched. Moreover, the bifurcation characteristics of load vertical vibration will be emphatically explored to reveal the instability mechanism of the HAGC system. In Section 2, the vertical vibration model of the load of the

HAGC system is established. In Section 3, the bifurcation characteristic of load vertical vibration is analyzed. In Section 4, the effects of some feature parameters on bifurcation behavior are thoroughly explored. In Section 5, some conclusions are provided.
