# The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System
## 1 Introduction  
The railway vehicle system operates in diverse geographical and climatic environments, encompassing plains, hills, mountainous areas, and plateaus. There are notable variations in temperature, humidity, altitude, sandstorms, rain, and snow along the line, which makes the operation conditions of the railway vehicle system complex [1]. The complex operating conditions may reduce the critical speed below the operational speed, resulting in hunting instability. Consequently, it is imperative to investigate further the potential consequences of railway vehicles operating at speeds exceeding the critical speed. Early studies on hunting stability were primarily conducted using linear models and subjected to a linear stability analysis [2; 3; 4; 5; 6]. The presence of nonlinear factors, such as wheel-rail contact relations, dry friction, flange forces, etc., makes the railway vehicle system a typical nonlinear and non-smooth system [7]. Based on the bifurcation theory, scholars have extensively focused on investigating the critical speed of railway vehicle systems. [PERSON] [8] investigated the condition that the equilibrium position converted toa periodic solution and analyzed the stability of the periodic solution based on the Hopf-Friedrichs bifurcation theory. [PERSON] et al. [9] conducted a comparison between the nonlinear and the linear critical speed. They emphasized the higher practical significance of the nonlinear critical velocity and analyzed the effect of suspended dampers on it. [PERSON] and [PERSON] [10] considered creep force and the flange contact nonlinearities in a single wheelset system and found that the nonlinearities of suspensions significantly affect the hunting motion and nonlinear critical velocity. [PERSON] et al. [11; 12] calculated the speed at which Hopf bifurcation occurs in the whole vehicle system based on a numerical simulation and verified it in a rolling test bed. [PERSON] [13] found that stable equilibrium points and periodic solutions coexist within a specific velocity interval when a wheelset system undergoes hunting instability via a subcritical Hopf bifurcation. [PERSON] and [PERSON] [14] found that the form of bifurcation that initiates hunting instability may be the supercritical or subcritical Hopf bifurcations due to differences in vehicle system parameters. [PERSON] and [PERSON] [15] investigated the influence of various wheel shapes on the type of Hopf bifurcation based on the enter manifold theorem. [PERSON] et al. [16] analyzed the Hopf bifurcation behavior for a single wheelset system for resonant and non-resonant conditions. They found that increasing the longitudinal suspension stiffness increases the vehicle system stability. Thus, it can be seen that based on the bifurcation theory, the mechanism of hunting instability and the associated bifurcation types could be effectively analyzed to identify the critical speed of the system. At the same time, the effects of parameter changes on the critical speed and bifurcation types could be analyzed. Exploring bifurcations, the evolution of periodic hunting motions at higher speed ranges and the analysis of wheel-rail impact characteristics have attracted the attention of many scholars. [PERSON] and [PERSON] [17] found stationary, periodic, and chaotic motions with a numerical simulation in lateral dynamic models of a single wheelset and bogie system, respectively. Subsequently, [PERSON] and co-workers performed a series of important works based on [PERSON]'s complex bogie and discovered its extremely diverse dynamic behavior, including multiple bifurcations and attractors [18; 19; 20; 21]. Bifurcation-induced asymmetric branching was found as well as the coexistence of periodic and chaotic motion behavior [18; 19]. [PERSON] and [PERSON] [20] analyzed the transit of hunting motion from equilibrium to chaotic motion in detail. They found high-order subharmonics in a very narrow parameter range, whereas the transit of periodic motion to chaotic motion initiated with Neimark-Sacker bifurcation.