In this paper, the dynamic model of the nose landing gear of a single-wheeled aircraft is investigated. After smooth fitting of the non-smooth term, the system is transformed into normal form by coordinate transformation. We calculated the first Lyapunov coefficient of the landing gear system by using the Hopf bifurcation theory, and judge the type of Hopf bifurcation according to its sign. The correctness of the theoretical derivation is verified by numerical simulation. And then, we applied linear feedback control to the system and analyzed the influence of control parameters on Hopf bifurcation behavior. A nonlinear cubic feedback controller is applied to the system, and the influence of the control parameters on the amplitude of the limit cycle is discussed through the amplitude calculation formula. The results show that the linear controller can make the Hopf bifurcation point of the system move back, thus reducing the unstable region of Hopf bifurcation. The nonlinear controller can reduce the amplitude of the limit cycle generated by Hopf bifurcation without changing the Hopf bifurcation point of the system. The results can provide some theoretical guidance for the structural optimization of aircraft nose landing gear system.