## 1 Introduction
[12] simulated a peloton applying experimental data in a points race and discussed the importance of cyclists' maximal sustainable outputs (MSO) in terms of power output, the proportion of the current power output to cyclists' MSOs in both the front and drafting positions, and how these factors affect peloton configurations. These factors are the principal control factors that determine peloton configurations. [PERSON] [13] refined the description of the stretched phase in terms of the mathematical model described by [PERSON] et al. [11; 12], that is, as occurring within the term of their PCR (peloton conversion ratio) equation. However, the dynamics of group separation and recombination in cycling races have not yet been analyzed in detail. Studies of bird flocks have used the function of the velocity of individuals as a variable to represent the population order, such as the polarization of the group's direction [5]. During velodrome cycling competitions, the direction of travel is determined in one direction. Therefore, the velocity of the cyclist running at the head of each group may reasonably be the representing variable of the movement of each group; the velocity of the cyclist who leads the whole and, if there is an escape group, the velocity of the leading cyclist of the main group, which is the largest group, would be the representation of the group state. Additionally, the relative speed difference between the leading cyclist and the leading cyclist in the main group is also considered a promising candidate for a descriptive variable affecting the state transition in the cycling race. Furthermore, the time difference between the escape group and the main group is represented by the group state. In previous research, [PERSON] et al. correlated the peloton phase to cyclists' power output [11]. The cyclists' MSO in the leading position and drafting position were determinant variables of phase states and their transitions. However, since MSO is unknown information during the racing competition, and the external power to overcome air resistance is proportional to the cube of speed [14], speed can be used as alternative information for MSO to describe the phase state and transition of bicycle racing. Therefore, we firstly hypothesize that the velocity of the leading cyclist, the velocity of the cyclist in the main group, their speed difference, and the time difference are the candidate descriptive variables for the state transition of groups in the cycling competition, wherein the groups separate and recombine. Hysteresis often presents in the change in order based on self-organization [15; 16]. Hysteresis is the dependence on the state of the system concerning history. In human movement, [PERSON] [17] found that a phase transition from anti-phase to in-phase occurs in the movement coordination within an individual; however, an in-phase to anti-phase transition could not occur, suggesting hysteresis. [PERSON] et al. [18] also showed that hysteresis occurs during the coordination task of leg movements between two individuals sitting on each other, switching between two modes (in-phase and anti-phase synchronization). [PERSON] et al. [19] verified the hysteresis in the phase transition of gait with four legs by using a robot. In the sports events, [PERSON] and [PERSON] [20] showed a third-order sequence effect due to hysteresis in the continuously repeated switching of the forehand and backhand tennis strokes. [PERSON], [PERSON], and [PERSON] [21] also showed that hysteresis occurs in selecting fore-backhand strokes for table tennis. This hysteresis occurs as two discrete processes measured separately from each other. Besides these examples, hysteresis can occur as an asymmetry in one continuous dynamical process, for example, vehicle traffic hysteresis [22]. [PERSON] examined hysteresis in a peloton applying the definition in vehicle traffic [23]. He identified three forms of hysteresis using flow analyses based on vehicle traffic research.