## 1 Introduction  
Piecewise-type nonlinearities such as multi-staged clutch dampers used in a practical torsional system induce highly complex dynamic responses. In the middle of these nonlinear dynamic behaviors, sub-harmonic responses are relatively difficult to detect by employing the harmonic balance method (HBM), the basic matrix of which is constructed by the integer-based, incremental formulations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. In addition, various modifications have been implemented to calculate sub-harmonic responses. For example, nonlinear frequency response calculations of a torsional system with clearance-type nonlinearity have been developed by employing a multiterm HBM [1, 2, 5, 8, 10, 12]. Our study suggested adapting the relevant time period for the sub-harmonic frequency values. [PERSON] et al. suggested an excitation perturbation method to investigate the sub-harmonic resonance [10]. To capture the sub-harmonic effects, the authors modified the input conditions. For example, the relevant sub-harmonic input terms were artificially included, which triggered relevant sub-harmonic responses. In addition, various prior studies have discussed nonlinear problems by employing the HBM with respect to super- and sub-harmonic responses [2, 7, 10, 12].  
Despite the successful determination of the sub-harmonic responses, the stability conditions of the HBM often cannot adequately explain the practical dynamic conditions.  
To understand the dynamic behaviors in complex stable and unstable conditions, bifurcation diagrams have been presented based on time-domain solutions. With regard to bifurcations, various approaches and results have been discussed. For instance, [PERSON] and [PERSON] investigated sub-harmonic and chaotic motions in a multi-mesh gear train by using a nonlinear time-varying dynamic model [5]. [PERSON] et al. examined bifurcation phenomena in a codimension-2 parameter space by using the HBM [14]. [PERSON] et al. suggested the original extended system model for the detection and tracking of bifurcations by using the HBM with an arc-length continuation technique [15].  
To the best of our knowledge based on the HBM and its relevant techniques, this study investigates the dynamic characteristics that occur in the sub-harmonic regimes, which are primarily concerned with a practical multi-staged clutch dampers that are employed in the conventional vehicle system. Thus, there are two specific objectives of this study. The first objective is to investigate the sub-harmonic regime which is just determined as unstable conditions based on the HBM. To capture the sub-harmonic resonances, the fictitious sub-harmonic input conditions will be employed numerically, which will advance the prior study [2]. The second objective is to examine the complex dynamic characteristics caused by a physical multi-staged clutch dampers by investigating the bifurcation diagrams that generally occur in the sub-harmonic unstable regimes. This will result in an increased understanding regarding the practical dynamic motions in a vehicle driveline system. In addition, this study focuses on one specific multi-staged clutch damper model in a torsional system with a single-degree-of-freedom (1 DOF) by limiting the sub-harmonic resonance areas around \(\varpi=1\). In addition, Broader range of dynamic responses and their analysis of the bifurcation characteristics in the same system as described in Figure 1 can be referred [20].