# Dynamic Investigations of Shared Bicycle Operators' Competition Based on Profit Maximization
## 4 Numerical Simulation
### System Bifurcation and Attractive Basin of Equilibrium Point  
Fix the system parameters, then the impact of the time cost, comfort loss cost, discount, and cost conversion coefficient of shared bike attractor operators \(T_{1}\) and \(T_{2}\) on the system, as illustrated in Figure 2. Figure 2 comprises six subplots, each illustrating the impact of different factors on the demand of the operator. Subplot (a) presents the impact of comfort loss cost \(\beta_{1}\) on the demand of operators \(T_{1}\). Subplot (b) presents the impact of the time cost \(k\) on the demand of operators \(T_{2}\). Subplot (c) presents the impact of the comfort loss cost \(\beta_{2}\) on the demand of operators \(T_{1}\). Subplot (d) presents the impact of travel distance \(D\) on the demand of operators \(T_{2}\). Subplot (e) presents the impact of cost conversion coefficient \(\mu_{2}\) on the demand of operators \(T_{1}\). Subplot (f) presents the impact of discounts \(S\) on the demand of operators \(T_{2}\). Figure 1: Stability region of the equilibrium point \(E_{4}\). Figure 2: \(Cont\). In Figure 2, the dynamic adjustment speed of shared bicycle operators is fixed at \(v_{1}=1.2\), \(v_{2}=1.3\), and the initial demand value of the operators is \((x_{1},x_{2})=(0.03,0.01)\). From subplot (a), one can see clearly with the increase in comfort loss cost \(\beta_{1}\), the demand for operator \(T_{1}\) changes from a period-doubling bifurcation and finally reaches a stable state in the region. Once the cost of comfort loss exceeds 0.4, the demand for operator \(T_{1}\) will decrease to zero, leading to operator \(T_{1}\)'s exit from the market. On the contrary, as the comfort loss cost \(\beta_{1}\) increases, the demand for operator \(T_{1}\) will shift from a chaotic state to a period-doubling bifurcation and eventually to a stable state, and the demand for operator \(T_{1}\) will gradually increase (see subplot (c)). Correspondingly, subplot (e) shows the increase in cost conversion coefficient \(\mu_{2}\) also causes the demand for operator \(T_{1}\) to change from a chaotic state to a period-doubling bifurcation, ultimately stabilizing at 0, resulting in operator \(T_{1}\) exiting the market. The results of subplot (d) indicate that as the distance increases, the demand for operator \(T_{2}\) stabilizes, despite the discount incentives offered by the operator. This is because longer distances make passengers more inclined to opt for public transportation such as buses or subways. Empirical evidence from various studies, as referenced in the literature [18; 36], demonstrates a negative correlation between the frequency of shared bicycle rentals and the distances traveled. Furthermore, the research presented in [19] underscores a strong correlation between the decline of short-distance public transit services within a 3 km radius and the increased utilization of shared bicycles. The increase in time cost \(k\) and travel distance also leads to a decrease in demand for operator \(T_{2}\), while the system also achieves a period-doubling bifurcation process with the time cost increases from 0.18 to 0.4 and distance increase from 0.4 to 1.2. When discounts \(S\) increase,  
Figure 2: (**a**) Bifurcation diagram of the demand for \(T_{1}\) when varying parameter \(\beta_{1}\). (**b**) Bifurcation diagram of the demand for \(T_{2}\) when varying parameter \(k\). (**c**) Bifurcation diagram of the demand for \(T_{1}\) when varying parameter \(\beta_{2}\). (**d**) Bifurcation diagram of the demand for \(T_{2}\) when varying parameter \(g\). (**e**) Bifurcation diagram of the demand for \(T_{1}\) when varying parameter \(mu_{2}\). (**f**) Bifurcation diagram of the demand for \(T_{2}\) when varying parameter \(S\). the demand for operator \(T_{2}\) eventually becomes chaotic (see subplot (f)). This indicates that substantial discounts can plunge the system into chaos, which is detrimental to market development. To maintain market competitiveness, discounts must remain within a specific range.