## 1 Introduction  
Delay differential equations (DDEs) often appear in various fields of science and engineering, such as control systems [1], lasers [2], neuroscience [3] and cutting process dynamics. For cutting dynamics, the cutting effect of tools can cause vibrations in the cutting system, resulting in accelerated wear of tools and influencing the cutting process in machines (turning, milling and grinding), coal seam mining, geological prospecting and oil drilling. [PERSON] [4] proposed the theory of regenerative cutting of the tool. During the cutting process, the cutting thickness of the workpiece is constantly changing due to the relative vibration of the tool and the workpiece. The cutting force of the cutter is a function of cutting thickness and depends on the tool's current and delay position. This theory has been widely accepted as the basis for cutting theory and experimental research. This paper mainly studies the non-linear dynamic stability of the deep hole drilling in coal seam gas exploitation (see Figure 1). Mine gas extraction is an important guarantee of safety and efficiency in coal mining. However, the drilling process can lead to strong self-excited vibration, which will affect the drilling stability. The self-excited vibration in drilling is similar to the chatter vibration in the cutting; both are caused by the regenerative cutting of the tool or drill bit. There are three types of self-excited vibration of drill string, axial vibration, torsional vibration and lateral vibration. They can lead to bit bounce, stick-slip and bit whirl, respectively [5]. These vibrations can affect the stability of machines and the drill string could be broken when the frequency of self-excited vibration is close to the natural frequency of the system. To avoid these vibrations and improve the drilling efficiency, the specific conditions need to be studied. [PERSON] [6,7] contrasted the bifurcation phenomenon of the stick-slip model, the whirl model, and the stick-slip whirl model at the equilibrium point of the drilling system in oil drilling and found that the whirl had a certain influence on the stick-slip, mainly due to the existence of fluid force in drilling. However, the effect of fluid force is not so obvious in directional drilling due to the different drilling's structures. [PERSON] [8, 9, 10] analyzed the influence of whirl on the frequency components and the motion law of the axis trajectory of the drilling system by establishing a drilling test bed. However, the research was more biased towards the rubbing of rotor and stator, and could not really reflect the cutting motion of the drill bit. In fact, the regenerative cutting motion of the bit is the main cause of the system vibration, which is determined by the axial feed and torsional speed of the drilling system. [PERSON] [11] used singular perturbation analysis to decouple axial and torsional dynamics in the model, and then conducted a detailed analysis of the axial limit cycle ensuring drilling outside of an unstable region. [PERSON] [12] considered the axial damping in Germany's model and analyzed the effect of axial periodic motion by a semi-analytical method. The results of the literature have confirmed that the influence of axial vibration on stick-slip vibration cannot be ignored. On the other hand, stick-slip was observed at low rotational velocity and high feed velocity of the bit [13,14], while bit whirl occurred at high rotational velocity and low feed velocity [15,16]. So, stick-slip vibration is the main form of self-excited vibrations for deep hole drilling with low rotational velocity and high feed velocity [17]. [PERSON] [18] and [PERSON] [17] specifically described stick-slip vibration that was characterized by alternate stick and slip phases. In a stick phase, the bit stops and the drill pipe continues to twist. In a slip phase, the rotational velocity of the bit increases instantly because of the energy stored by the drill pipe in the stick phase.