## 3 Linear Stability Analysis
In order to make a more intuitive analysis of the change law of the critical cutting depth with the rotation speed of the SDD and CD model, 3D waterfalls (Figure 6) were drawn. These figures are the change law of the critical non-dimensional cutting depth ratio of the first lobe and the seventh lobe, respectively. The \(Z\)-axis is the critical non-dimensional cutting depth ratio between the SDD and CD model (\(\psi_{SDD}/\psi_{CD}\)), the \(X\)-axis is the control parameter \(\rho\), and the \(Y\)-axis is the vibrational frequency \(\omega_{c}\). From the figure, it can be seen that the critical non-dimensional cutting depth ratio increases with the increase of control parameter \(\rho\) and vibrational frequency \(\omega_{c}\), the maximum can reach 1.1. Comparing the two figures, the change law of the critical non-dimensional cutting depth ratio is consistent in each lobe and there is no significant difference. On the other hand, the ratio is  
Figure 5: Stability charts of SDD and CD model. small (closer to 1) in the case of small control parameters, but the value of critical non-dimensional depth ratio increases significantly with the variation of vibrational frequency \(\omega_{c}\). In order to verify the accuracy of the stability lobes diagram, three points \(A\), \(B\) and \(C\) at the \(\Omega=2\) on the lobe graph are chosen. The coordinates are \(A\) (2, 0.01), \(B\) (2, 0.03) and \(C\) (2, 0.0234), respectively. \(A\) is located in the stable cutting region, with \(B\) in the unstable cutting region. For SDD model, point \(C\) is located at the stability boundary when \(\rho=11\). The perturbation axial velocity and perturbation torsional velocity at \(A\), \(B\) and \(C\) of the two models were solved by the Runge-Kutta numerical algorithm, as shown in Figures 7-9, where the parameter value is \(v=0.1\), \(k_{r}=10\), \(Q_{A}=Q_{T}=5\), \(N=4\). At point \(A\), the responses of the two models are significantly attenuated. There is no significant difference in axial velocity between the two models. However, the SDD model has greater torsional speed amplitude because the state-dependent delay depends not only on the time but also on the system's torsional state. At point \(B\), the torsional speed of the two models increases dramatically with time and the system is extremely unstable. When the disturbance torsional speed is equal to the amplitude of the system input rotational speed, the system enters the stick-slip state. At point \(C\), the SDD model developed into periodic motion after 4 s, and the axial and torsional phase diagrams show a strict limit cycle state, as seen in Figure 10.  
The above numerical results show that the stability lobes diagram can accurately predict the stable and unstable cutting area of the drilling system at \(\Omega=2\). However, at a certain speed, this prediction is less accurate due to the existence of Hopf bifurcation. Therefore, the Hopf bifurcation area of the drilling system is solved in this paper. Figure 6: The change law of the critical dimensionless cutting depth ratio of SDD and CD model: (**a**) the first lobe; (**b**) the seventh lobe. Figure 7: Evolution with time of responses for the CD model and SDD model at point \(A\).