## 1 Introduction
This phenomenon would periodically change stress and strain to accelerate the fatigue failure of the drilling system. The most common stick-slip model used in the present study is non-linear, including material non-linearity [19] as well as structural non-linearity [20]. The effect of cutting force is the largest among all the non-linear factors [21,22]. [PERSON] and [PERSON] [23] proposed a non-linear cutting model that included non-linear stiffness and non-linear cutting force. [PERSON] [17] considered non-linear contact stress in the drilling process. These studies have shown that the non-linear factors have a significant impact on system stability. [PERSON] and [PERSON] [24] pointed out the non-linearity of a drilling system is mainly caused by the non-linear cutting force. Their study was supported by subsequent experimental research. The factors of the non-linear cutting force are mainly reflected in the changing delay time. However, the most commonly delay model used in previous studies is the linear constant delay (CD) model, which is equal to the period of the drill bit (workpiece) rotation in drilling (turning, milling). The mathematical model is usually an autonomous delayed differential equation with constant delay (CD-DDE). Models with constant time delay capture the main characteristics of  
Figure 1: Deep hole drilling in coal seam gas exploitation: (**a**) Deep drill rig; (**b**) 3D model. - regenerative dynamics, but these models can only be applied in linearized systems. When the system is non-linear, the calculated results cannot be in good agreement with the experiments. Recently, the concept of non-linear state-dependent delay (SDD) was presented [25; 26]. [PERSON] [27] and [PERSON] [28] pointed out that the regenerative delay was time-dependent due to the feed rate in an accurate model of the milling process. They showed that the traditional constant delay models overestimate stability limits for large feed rates. The regeneration of delay in the turning system is mainly determined by the workpiece's speed, but also affected by the current and delay position of the tool. This is the state-dependency delay (SDD) model, and the mathematical model is a delayed differential equation with state-dependent delay (SD-DDE). The SD-DDE is an autonomous non-linear equation and its solution is non-differentiable due to the state-dependent time delay \(\tau(x(t))=\tau_{0}+x(t)\), as shown in Equation (1). CD-DDE can be considered a linear variational system, corresponding to Equation (1) at the equilibrium point \(x(t)=0\), as shown in Equation (2). Therefore, the analysis of a drilling system represented by a SD-DDE is more complicated than the conventional constant DDE (C-DDE). [PERSON] [29; 30] investigated drilling with drag bits and found that state-dependent regenerative delay arose due to the torsional vibration of the tool. For deep hole drilling, the effect of state-dependent delay is especially apparent. \[\dot{x}(t)=x(t-(\tau_{0}+x(t))) \tag{1}\]  
\[\dot{y}(t)=y(t-\tau_{0}) \tag{2}\]  
In order to study the influence of non-linear state-dependent delay on deep hole drilling systems, this paper studied and compared the stability of CD and SDD models through the lobes diagram. On the other hand, control parameters (rotational velocity and feed velocity) could be chosen to ensure a stable cutting process without stick-slip vibration based on the lobes diagram. The non-enclosed stability lobes diagram is a function of axial cutting depth on spindle speed, which can be drawn using numerical or semi-analytic methods [31; 32; 33; 34; 35]. Recently, [PERSON] and [PERSON] [36; 37] carried out a stability analysis using a semi-discretization and constructed a stability volume in the three-dimensional parameter space of spin speed, cutting depth and a cutting coefficient. They used a two degree-of-freedom model and a multisegment model of drill strings with state-dependent delay. [PERSON] [38] plotted the stability charts in the plane of drilling rates and rotary speeds.