# The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System
## 1 Introduction
[PERSON] [21] emphasized the dependence of critical velocity on the initial state and analyzed the influence of parameter changes. [PERSON] [22; 23; 24; 25] used consequence bifurcation diagrams to investigate asymmetric behavior in symmetric railway systems. [PERSON] [26] studied stable and unstable solutions near the critical velocity and analyzed the parameter sensitivity. [PERSON] and [PERSON] [27; 28] extended the bifurcation theory of railway vehicle systems in straight rails to research curved tracks. It was found that there are similar mechanisms of hunting instability and complex types of hunting motion and bifurcation forms in curved tracks. In addition, some studies have found that the primary hunting motion of vehicle systems at low speeds occurs during the speed-up of the vehicle system, which seriously affects passenger comfort [29; 30; 31]. Based on the above analysis, there are complex periodic motion patterns as well as bifurcation types in the speed range above the critical speed. The nonlinear factors in the hunting motion mainly arise from the wheel-rail creep-slip forces and the wheel-rail impacts. In studying the cone-tread wheel and small-amplitude hunting instability, wheel-rail contact geometry relationships based on equivalent conicity linearization are widely used [32; 33; 34; 35; 36]. In studies using worn-type wheels, scholars have used polynomial fitting [37; 38], least-squares fitting [39], measured data [12], and software simulations [40; 41; 42] to obtain nonlinear wheel-rail contact geometry relationships. In severe hunting instability, when the wheelset lateral displacement exceeds the wheel-rail clearance, it will be subjected to flange forces. A description of the flange force as a piecewise linear spring force is widely used [21; 43; 44], which makes the vector field of flange forces not differentiable at the inflection point and makes the system a typical non smooth system. Polynomial fitting is often used to avoid non-smooth properties, which enables the utilization of various theories and methods suitable for smooth nonlinear systems [38; 45]. However, it also has disadvantages, such as the widely used Wagner's function is a five-order polynomial fitting [13]. Still, the direction of flange force at small displacement is opposite to the actual direction. Therefore, in the study of hunting motion with larger lateral displacements, it is crucial to fully consider the non-smooth and nonlinear characteristics of wheel-rail contact relations and wheel-rail impacts. So far, the study of the hunting motion of railway vehicle systems based on the bifurcation theory mainly adopts single-parameter bifurcation, and a few two-parameter analyses have only obtained Hopf bifurcation lines in the parameter plane [46; 47], which fails to effectively identify the mode types of periodic hunting motion. In this paper, the nonlinear, non-smooth wheel-rail contact relation and the effect of flange force are fully considered. The pattern types of periodic hunting motion are effectively identified in the two-parameter plane, their diversity and bifurcation characteristics are analyzed, and their wheel-rail impact characteristics are investigated. The paper is organized as follows. The lateral dynamic mode of a railway bogie system is presented in Section 2. In Section 3, the Poincare sections are constructed, and the two-parameter co-simulation theory is briefly introduced. In Section 4, lateral dynamic characteristics of the bogie system are numerically analyzed. Based on the two-parameter co-simulation, the mode types and distribution laws of the periodic hunting motion in the (\(v\), \(\mu\))-plane and the wheel-rail impact characteristics are studied in Section 4.1. Mechanisms of hunting instability, bifurcation, and transition laws of periodic hunting motion are studied in Section 4.2. Conclusions are drawn in Section 5.