# The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System
## 4 Numerical Results and Discussions
### Bifurcation and Evolution of Periodic Hunting Motion
#### 4.2.1 Select \(\mu\) = 0.12  
Figure 8 shows the bogie system's bifurcation diagrams that laterally cross the (\(v\), \(\mu\))-plane at \(\mu\) = 0.12. Figure 8a is the periodic bifurcation diagram based on the period mapping section \(\Sigma_{1}\). The bifurcation diagram of wheelset-1, based on the wheel-rail impact mapping section \(\Sigma_{w1}\), is depicted in Figure 8b. Meanwhile, the bifurcation diagram of wheelset-2, based on the wheel-rail impact mapping section \(\Sigma_{w2}\), is depicted in Figure 8c. Through numerical simulations, it can be obtained that the critical speed is \(v_{1}\) = 78.8 m/s, the velocity at which wheel-rail impact occurs in wheelset-1 is \(v_{2}\) = 83.45 m/s, and the velocity at which wheel-rail impact occurs in wheelset-2 is \(v_{3}\) = 89.46 m/s. When the operating velocity \(v\) is lower than the critical velocity \(v_{1}\), the system is in stable hunting motion and can converge to the stationary solution after external disturbance. Figure 9a,b present the time series and phase portraits of the system converging to the equilibrium point after being subjected to equal external excitations for \(v\) = 70 m/s and \(v\) = 78 m/s, respectively. It can be found that the convergence of the system slows down significantly when approaching the critical velocity. As \(v\) = \(v_{1}\) = 78.8 m/s, the system suffered hunting instability via the Hopf bifurcation, after which the system could not converge to equilibrium after the excitation. Instead, it is in periodic hunting motion. When the running speed slightly exceeds the critical speed in the interval \(v_{1}\) < \(v\) \(\leq\)\(v_{2}\), the system is in a wheel-rail impact-free hunting motion (1-0-0), as shown in Figure 10a,b. When \(v\) = \(v_{2}\) = 83.45 m/s, wheelset-1 is in 1-0-0 motion with grazing, as depicted in Figure 10c. Grazing motion is a motion state in which the velocity is infinitely close to zero when the displacement reaches the constraint boundary, and it is a typical non-smooth bifurcation that exists only in non-smooth systems. After grazing bifurcation, wheelset-1 exhibits the 1-1-1 motion in Figure 10d, while wheelset-2 experiences delayed wheel-rail impact compared to wheelset-1 and maintains the 1-0-0 motion, as illustrated in Figure 10d,e. At \(v\) = \(v_{3}\) = 89.46 m/s, wheelset-2 exhibits the 1-0-0  
Figure 7: Three-dimensional cloud map and projection of contour map of maximum wheel–rail impact velocities. (**a**) wheelset-1; (**b**) wheelset-2. motion with grazing, as depicted in Figure 10f. After that, both wheelsets are in 1-1-1 motion involving wheel-rail impacts; see Figure 10g,h. Figure 8: Bifurcation diagrams that laterally cross the (\(v\), \(\mu\))-plane at \(\mu=0.12\). (**a**) The period mapping section \(\Sigma_{1}\); (**b**) the wheel–rail impact mapping section \(\Sigma_{w1}\) of wheel-1; (**c**) the wheelset–rail impact mapping section \(\Sigma_{w2}\) of wheel-2. (The red and blue bifurcation branches correspond to right and left side wheel–rail impacts, respectively). Figure 9: Time series and phase portraits of wheelset-1 and wheelset-2. (**a–c**) \(v=70\) m/s; (**d–f**) \(v=78\) m/s. As \(v\) increases, it is accompanied by more complex periodic motion patterns and multiple bifurcation types. When \(v=95.6\) m/s, the system suffers the Neimark-Sacker bifurcation, and the two wheelsets go from the 1-1-1 motion to the quasi-period 1-1-1 motion. Figure 11a,b shows the phase diagrams of the two wheelsets at \(v=96\) m/s. The phase trajectory of quasi-periodic motion exhibits a closed black band. When \(v=97.25\) m/s, the quasi-period 1-1-1 motion of wheelset-1 grazes with the rail, as seen in Figure 11c, and wheelset-2 is in quasi-period 1-1-1 motion, as seen in Figure 11d. After G Bifurcation, wheelset-1 enters chaotic motion with two attractors, and wheelset-2 enters chaotic motion with one attractor; see Figure 11e,f.