## 1 Introduction  
There are many phenomena in different fields of science and engineering where the physical response of a variable involves not only the value at time \(t\) but also the effects that occur in an earlier state \(t-\tau\). Thus, delay systems appear in many engineering problems, such as in the shimmy effect (wheel vibration) [1], vehicle traffic models [2], feedback stabilization problems [3] and in the regenerative vibration of machine-tools better known as chatter [4]. In cases where the net force depends on the current values and some past values (history) such as position and speed, the system dynamic behavior can be modeled using a differential delay equation (DDE). It is well-known that during a milling process, unstable vibrations also known as self-excited vibration or chatter may occur. Chatter reduces the machining efficiency due to low material removal rate by reducing the workload and affects surface quality, shortens tool life and accelerates tool wear. Researchers are studying several ways to overcome this limitation. [PERSON] et al. [5] studied the incorporation of a chatter detection system based on multiple sensors to milling operations for industrial conditions, [PERSON] et al. [6] used a method based on fractal dimension for the flank milling of a thin-walled blade, which can reflect the chatter severity level through the morphological change in signal. [PERSON] and [PERSON] [7], to mitigate chatter, presented an active controller based on the technique of discrete time sliding mode control (DSMC) blended with the type-2 fuzzy logic system. Moreover, [PERSON] et al. [8] presented a method based on a dynamic cutting force simulation model and a machine learning approach based on statistical learning theory to predict and avoid the cutting chatter. In addition, to control and suppress chatter vibrations, the use of piezoelectric actuators embedded in the tool holder [9], electromagnetic actuators integrated into the spindle system [10] and tunable clamping table [11] has been analyzed. In the milling process, the use of variable pitch cutters has demonstrated to improve productivity [12]. Different from the uniform pitch cutter, when a variable pitch cutter is used the dynamics model of cutting vibration changes from DDEs with a single delay to DDEs with multiple delays [13]. A common technique offline to predict unstable vibrations is the so-called stability lobes of the DDE based on Floquet theory [14], in which a curve describes the limit of stable vibration under feasible range values of cutting parameters. The stability analysis of the milling process with multiple delays has been studied through different methods. Among all these methods, those with variable pitch tools play a critically important role [15]. [PERSON] [16] was the one who first demonstrated the effectiveness of variable pitch cutters in suppressing vibrations in the milling process, he assumed a rectilinear tool motion for cutting teeth, and applied the theory of orthogonal stability to the irregular pitch of the tooth, by assuming an alternating step variation then, he obtained an expression of the stability limit as a function of the step angle variation. [PERSON] [17,18] proposed an analytical method for nonconstant pitch milling cutters from a design point of view, showing for some applications how this variable effect helps to reduce self-excited vibrations, so he found that chatter stability can be improved significantly even at slow cutting speeds by properly designing the pitch angles. [PERSON] et al. [19] used the frequency domain method to analyze the milling stability of the variable pitch cutter and introduced a method to select the optimal pitch angles. [PERSON] and [PERSON] proposed a mathematical approach, the cluster treatment of characteristic roots (CTCR), which optimizes the design of variable pitch cutters [20]. [PERSON] et al.