# The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System
## 4 Numerical Results and Discussions  
Hunting Motion Patterns and Distribution Regions in the Two-Parameter Plane and Wheel-Rail Impact Characteristics  
The running speed \(v\) is always changed during the operation of a railway vehicle. The railway vehicle service environment is complex, and the temperature, humidity, and wheel-rail medium changes (such as rain, snow, mud, oil, etc.) may make the wheel-rail adhesion coefficient \(\mu\) change. The select operating speed \(v\) and wheel-rail adhesion coefficient \(\mu\) construct the (\(v\), \(\mu\))-plane, where \(70\leq v\leq 170\) and \(0.1\leq\mu\leq 0.2\). In the numerical simulation, the system is given an initial value \(\mathbf{\dot{Y}}^{\mathrm{T}}=\left\{y_{w1},\dot{y}_{w1},y_{w2},\dot{y}_{w2},\dot{y }_{w1},\dot{y}_{w1},\dot{y}_{w2},\dot{y}_{w2},\ \ y_{t},\dot{y}_{t},\dot{y}_{t},\dot{y}_{t}, \dot{y}_{t}\right\}^{\mathrm{T}}=\left\{0.009,0.04,0,0,0,0,0,0,0,0,0,0,0,0\right\}^ {\mathrm{T}}\), and the entire parameter plane is scanned line by line with an incremental change in speed \(v\) starting from the lower left corner of the entire parameter plane at (\(v\), \(\mu\)) = (\(70\), \(0.1\)). Based on the computational method of two-parameter co-simulation introduced in Section 3 and the effective identification of the number of cycles and the number of wheel-rail impacts for the hunting motion, the pattern types and existence regions of the hunting motion are obtained in the (\(v\), \(\mu\))-planes associated with wheelset-1 and wheelset-2, respectively, as shown in Figure 4. Stable hunting motion (i.e., hunting motion that can converge to the equilibrium point after being disturbed) is expressed by SHM. The periodic hunting motion is represented by \(n\)-\(p\)-\(q\) (\(n=1\), \(2\), \(3\), \(\ldots\); \(p=0\), \(1\), \(2\), \(3\), \(\ldots\); \(q=0\), \(1\), \(2\), \(3\), \(\ldots\). ); in particular, 1-0-0 represents the periodic hunting motion without wheel-rail impact. The gray area mainly includes chaotic motion, quasi-periodic motion, and periodic hunting motion with minimal area. Partial bifurcation lines are marked in the (\(v\), \(\mu\))-plane, where 'Hopf Bif' represents Hopf bifurcation and 'G Bif' represents grazing bifurcation. The shortened representation also includes the following: 'SN Bif' represents saddle-node bifurcation, 'PD Bif' represents period-doubling bifurcation, 'N-S Bif' represents Neimark-Sacker bifurcation, and 'PF Bif' represents pitchfork bifurcation. Figures 5 and 6 show the wheel-rail impact velocity bifurcation diagrams transversely across the (\(v\), \(\mu\))-plane for wheelset-1 and wheelset-2, respectively. In Figure 4, the existence region of SHM (marked in red) is vertically banded and widens with the increase in \(\mu\), indicating that the bogie system's critical speed increases with \(\mu\). After that, Hopf bifurcation initiates hunting instability with increasing velocity, and the system enters the hunting motion without wheel-rail impacts (1-0-0). In the same parameter regions, wheelset-1 and wheelset-2 have the same number of periods (the system components coupled to each other have the same number of periods). However, the number of wheel-rail impacts shows significant diversity. For wheelset-1, as seen in Figure 4a, the existence region of the wheel-rail impactless motion 1-0-0 is distributed in a vertical band and narrows significantly with the increase in \(\mu\). Notably, 1-0-0 motion is crucial for monitoring hunting stability. Crossing the grazing (G) bifurcation line, the 1-0-0 motion transfers to the 1-1-1 motion. The existence region of 1-1-1 motion extends towards the higher velocity region at smaller \(\mu\), shows vertical banding in the larger \(\mu\) region, and narrows with increasing \(\mu\). For smaller \(\mu\), the 1-1-1 motion enters chaos directly, as seen in Figure 5a,c, or enters quasi-periodic motion via the Neimark-Sacker bifurcation and then transits to chaotic motion, as seen in Figure 5b. Chaotic and quasi-periodic motions form unidentified grey areas in the larger velocity region.