# 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
#### 4.2.2 Select \(\mu=0.18\)  
The bifurcation diagrams of the bogic system that laterally crosses the (\(v\), \(\mu\))-plane at \(\mu=0.18\) are shown in Figure 14, in which \(v_{1}=78.8\) m/s, \(v_{2}=82.7\) m/s, and \(v_{3}=86.4\) m/s. Compared to the one-parameter bifurcation diagram in Figure 8 at \(\mu=0.12\), for low \(v\), the most remarkable difference is that both wheelsets undergo two sequential G bifurcations, thus entering the 1-2-2 motion. When \(v_{1}<v\leq v_{2}\), wheelset-1 is in the 1-0-0 motion. When \(v=v_{2}=82.7\) m/s, it enters 1-1-1 motion through the first G bifurcation. When \(v=92.55\) m/s, it enters 1-2-2 motion through the second G bifurcation. The phase diagram of the transition process is shown in Figure 15a-e. Similarly, Figure 16a-e describes the transmigration for wheelset-2 from the 1-0-0 to the 1-2-2 motion via two G bifurcations. The first G bifurcation occurs at \(v=v_{3}=86.4\) m/s, and the second G bifurcation occurs at \(v=v_{3}=92.6\) m/s. It can be seen that wheelset-2 undergoes G bifurcation later than wheelset-1. It can be observed that the 1-2-2 motions of two wheelsets show asymmetric branches and amplitude jumps in the medium velocity interval. Figure 17 presents the combined bifurcation diagrams in the velocity interval \(v\in[110,\,126]\) m/s, and the black and magenta bifurcation branches correspond to the numerical simulation results for increasing (\(vZ\)) and decreasing (\(v\)b) velocities, respectively. The different-colored bifurcation diagram branches between pitchfork (PF) bifurcation point A and jump (J) bifurcation point B are caused by the asymmetry caused by PF bifurcation, which are coexisting periodic solutions under different initial conditions. Figure 18(a1,a2) shows the phase diagram of coexisting periodic solutions of 1-2 motion (in black) and 1-2 motion (in magenta) of two wheelsets at \(v=113\) m/s. It can be observed that the two coexisting periodic solutions are symmetric about the center of the coordinate origin. Between J bifurcation point B and J bifurcation point C, there is a coexistence of periodic motions due to amplitude jumps initiated by J bifurcation. For wheelset-1, J bifurcation only leads to amplitude jumps, and the motion mode remains 1-2-2 motion. Figure 18(b1) shows the coexistence of 1-2-2 motion (in black) and 1-2-2 motion (in magenta) at \(v=\)122 m/s. For wheelset-2, as the speed increases, 1-2-2 motion jumps to 1-1-1 motion. When the velocity decreases, 1-1-1 motion jumps to 1-2-2 motion. Thus, in this velocity interval, 1-1-1 motion (in black) and 1-2-2 motion (in magenta) coexist; see Figure 18(b2). Importantly, PF bifurcation or G bifurcation leads to the coexistence of multiple period motions. The different initial states will cause the system to have different periodic motion patterns. The constant change in environmental excitations in actual working conditions may cause the system to switch repeatedly between multiple coexisting motions, leading to abnormal vibration, increased wear, and so on. Figure 13: Phase portraits of two wheelsets. (**a1**) \(v=\)135 m/s, wheelset-1, 1-2-2 motion; (**a2**) \(v=\)135 m/s, wheelset-2, 1-1-1 motion; (**b1**) \(v=\)170 m/s, wheelset-1, quasi-period 1-2-2 motion; (**b2**) \(v=\)170 m/s, wheelset-2, quasi-period 1-1-1 motion. Figure 14: Bifurcation diagrams that laterally cross the (\(v\), \(\mu\))-plane at \(\mu\) = 0.18. (**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. Figure 15: Phase portraits of wheelset-1 transit from 1-0-0 to 1-2-2 motion by two successive G bifurcations. (**a**) \(v\) =81 m/s, 1-0-0 motion; (**b**) \(v\) =82.7 m/s, 1-0-0 motion with grazing; (**c**) \(v\) =84 m/s, 1-1-1 motion; (**d**) \(v\) = 92.55 m/s, 1-1-1 motion with grazing; (**e**) \(v\) = 94 m/s, 1-2-2 motion. Figure 16: Phase portraits of wheelset-2 transit from 1-0-0 to 1-2-2 motion by two successive G-bifurcations.