With the rise of the sharing economy, shared bicycles have become an important component of urban transportation. This paper explores the nonlinear dual oligopoly system for the Cournot model in the bike-sharing market; both operators have maximized profits as their competitive goals. The analysis of pivotal factors influencing passenger preferences, including pricing discounts and comfort levels, is meticulously depicted by a bifurcation diagram. A new chaotic attractor—the shared bicycle attractor—is discovered. The research results indicate that larger discounts and adjustment speeds can cause the system to be in a chaotic state, which is not conducive to the long-term development of operators, although discounts can indeed attract more passengers to a certain extent. On the other hand, the increase in the marginal cost of comfort loss can also make it difficult for enterprises to operate, which requires continuous technological innovation to improve the comfort of cycling.

This paper investigates the competition between two types of bike-sharing services, particularly at bus stops, subway stations, and residential areas. Two types of shared bicycle travel choice models are constructed. A shared bicycle operator attracts users by implementing discount incentives, and the comfort levels of riding the two types of shared bicycles are different. The equilibrium fares, potential user demand, and operator profits under joint profit maximization, price competition, and potential user demand competition scenarios are derived, and the competitive results under the three scenarios are compared. The results show that, in the potential user demand competition, the difference in potential demand between the two operators is largest; in the joint profit maximization scenario involving shared bicycle operators, the difference in potential user demand is smallest. In all competitive scenarios, higher operating costs and costs in lowering comfort loss for the shared bicycle operators will increase fares; the substitution level between the two types of shared bicycles has a positive impact on potential user demand, and the higher the substitution level, the better the effect of discounts in attracting users.