## 5 Hopf Bifurcation Analysis
### Hopf Bifurcation for State-Dependent Delay Model  
Figure 16 shows the change of the Hopf bifurcation to the stability lobes of the state-dependent delay model with different values of control parameter \(\rho\). The blue line represents the subcritical Hopf bifurcation area and the red line represents the supercritical Hopf bifurcation area. In Figure 16, Hopf bifurcation does not change the entire stability lobes. It can be seen that the right side of the lobes becomes supercritical with the increase of \(\rho\), while the left side does not change. The drilling system has only supercritical Hopf bifurcation in the low rotational speed region (\(\Omega<0.5\)). In the subcritical Hopf bifurcation section, there is a stable periodic motion when drilling in a linear unstable region and the system's linear stability region does not have unstable periodic motion. This means that stick-slip vibration will not occur in the system when the stability parameter axis cutting depth \(\psi_{\mathrm{C}}\) is over a certain threshold. The periodic vibration occurs in the drilling system only when it loses stability, and the amplitude of stick-slip vibration increases with \(\psi_{\mathrm{C}}\). Therefore, in the supercritical Hopf bifurcation section, the system will not become unstable. Under practical conditions, the supercritical Hopf bifurcation is more advantageous than the subcritical Hopf bifurcation. Figure 14: The variation of the amplitude \(a\) with the dimensionless cutting depth \(\psi_{\mathrm{CD}}\) (\(\Omega=2\), \(\rho=11\)). Figure 13: The variation of the amplitude \(a\) with the dimensionless cutting depth \(\psi_{\mathrm{CD}}\) (\(\Omega=1.2\), \(\rho=11\)). Figure 15: The variation of the amplitude \(a\) with the dimensionless cutting depth \(\psi_{\mathrm{CD}}\) (\(\Omega=4\), \(\rho=11\)). Figure 17 shows the change of the control parameters \(\rho\) to the Hopf bifurcation properties (a) and the critical control parameters \(\rho\) of the Hopf bifurcation at different rotational speeds \(\Omega\) (b) in the first stability lobe (\(1<\Omega<4.5\)). In order to study the limit of deep hole drilling, we extend the range of \(\rho\) to twice the rated working condition (\(\rho\leq 70\)). It can be seen from the figure that no matter how \(\rho\) changes, the bifurcation properties of the drilling system will not change in the non-dimensional rotational speed range 1-3.5. Only when the non-dimensional rotational speed is greater than 3.5 will the subcritical Hopf bifurcation be converted into the supercritical Hopf with the increase in \(\rho\). In actual drilling, the corresponding control parameters can be selected according to Figure 17, and the system is in the zone of supercritical Hopf bifurcation, which can ensure the stability of the drilling system. To verify the Hopf bifurcation feature of the state-dependent delay model, the variation of amplitude \(a\) and axial dimensionless cutting depth \(\psi_{SDD}\) were plotted at \(\Omega=1.2\), \(2\), \(4\); \(\rho=11\) (see Figures 18-20). It can be seen from Figure 19 that the amplitude \(a\) is increasing in the stable region when the \(\psi_{c}\) passes the threshold. So the system is in the subcritical Hopf bifurcation section. In Figures 18 and 20, the amplitude \(a\) is increasing in the unstable region when \(\psi_{c}\) is above the threshold and the system is in the supercritical Hopf bifurcation section. Figure 16: The change of Hopf bifurcation to stability lobes of the SDD model: (**a**) \(\rho=11\); (**b**) \(\rho=30\). Figure 17: Change of criticality along the first stability lobe of SDD model (\(1<\Omega<4.5\)): (**a**) Change of the control parameters \(\rho\) to the Hopf bifurcation properties; (**b**) change of the critical control parameters \(\rho\) of the Hopf bifurcation at different rotational speeds \(\Omega\). The results of the Hopf bifurcation of the CD model and the SDD model are compared.