# 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
Afterward, the two wheelsets degenerate from chaotic to 1-2-2 motion and then enter quasi-period motion again through Neimark-Sacker bifurcation, followed by chaotic motion over a more extended velocity range. It can be observed that there are many subharmonic motion windows embedded in the chaotic motion velocity intervals. We focus on analyzing the properties of the subharmonic motion windows embedded in the velocity interval \(v\in[110,126]\) m/s. Figure 12 shows the combined bifurcation diagrams obtained with the numerical simulation with increasing and decreasing running speeds, respectively (the black and magenta bifurcation branches correspond to the numerical simulation results for increasing(\(vZ\)) and decreasing(\(vb\)) velocities, respectively). The objective is to further reveal the possible coexistence behavior of periodic hunting motions. It can be observed that there are two subharmonic motion windows in the velocity range \(v\in[110,126]\) m/s. In the first window of subharmonic motion at a low speed, the number of periods of the system is \(n=9\), which can be obtained from the number of branches of the period bifurcation diagrams shown in Figure 12a; wheelset-1 is in subharmonic hunting motion 9-18-18, wheelset-2 is in subharmonic hunting motion 9-13-13, and the number of wheel-rail impacts can be obtained from the branch numbers of the impact velocity bifurcation diagrams in Figures 12b and 12c, respectively. It should be emphasized that the subharmonic motions in this periodic window are symmetric, so there is no coexistence of periodic motions. The hunting motion patterns in the other subharmonic motion window in the higher velocity range are more complex, containing asymmetric subharmonic motions and coexistence behavior due to initial value sensitivity. Figure 10: Phase portraits of two wheelsets. (**a,b**) \(v=80\)m/s, wheelset-1 and wheelset-2, 1-0-0 motion; (**c**) \(v=83.45\) m/s, wheelset-1, 1-0-0 motion with grazing; (**d**) \(v=85\) m/s, wheelset-1, 1-1-1 motion; (**e**) \(v=85\) m/s, wheelset-2, 1-0-0 motion; (**f**) \(v=89.46\) m/s, wheelset-2, 1-0-0 motion with grazing; (**g,h**) \(v=92\) m/s, wheelset-1 and wheelset-2, 1-1-1 motion. ity. In Figure 12a, it can be observed that the number of periods of subharmonic motion generated by the period-doubling bifurcation is \(n=4\), \(n=8\), and \(n=16\), and the black bifurcation branches (numerical simulation with \(vZ\)) and magenta bifurcation branches (numerical simulation with \(v\)b) coexist. For wheelset-1, it can be observed in Figure 12b that periodic doubling bifurcation sequences generate subharmonic motions (4-8-8, 8-16-16, and 16-32-23). As seen in Figure 12c, for wheelset-2, there are not only subharmonic motions (4-6-6, 8-12-12, and 16-24-24) generated by periodic doubling bifurcation sequences. At the same time, there are more complex motion patterns caused by G and SN bifurcations. Figure 11: Phase portraits. (**a,b**) \(v=96\) m/s, quasi-period 1-1-1 motion of two wheelsets; (**c,d**) \(v=97.25\) m/s, quasi-period 1-1-1 motion of wheelset-1 grazing with rail and quasi-period 1-1-1 motion of wheelset-2; (**e,f**) \(v=97.5\) m/s, chaos motion of two wheelsets. Figure 12: Combined bifurcation diagrams in the speed range \(v\in[110,\,126]\) m/s. (**a**) The period mapping section \(\Sigma_{1}\); (**b**) the wheelset–rail impact mapping section \(\Sigma_{v1}\); (**c**) the wheelset–rail impact mapping section \(\Sigma_{v2}\). (The black and magenta bifurcation branches correspond to the numerical simulation results for increasing(\(vZ\)) and decreasing(\(v\)b) velocities, respectively). After exiting the velocity range of chaotic motion, the two wheelsets enter the 1-2-2 motion, as shown in Figure 13(a1,a2). As the speed increases, the 1-2-2 motion transitions from Neimark-Sacker bifurcation to the quasi-periodic motion, and the number of quasi-periodic motion attractors remains consistent with the number of periodic attractors; see in Figure 13(b1,b2).