Based on a stiffness system consisting of an air spring and a negative stiffness spring connected in parallel, a single-degree-of-freedom 1/4 vehicle suspension model was established. The continuation shooting method was applied to track the periodic solutions of the suspension system, and their stability was determined by the Floquet theory. The basin of the system were depicted using the principle of cell mapping, and the evolution of the basin under different parameters was analyzed. The results show that the suspension system exhibits local bifurcation behaviors such as saddle-node bifurcation and period-doubling bifurcation under periodic excitation. The study reveals that the transition from periodic solutions to chaos in the system is mainly associated with Ι-type intermittent related to saddle-node bifurcation . When the unstable periodic orbit collides with the chaotic attractor, the system will produce dynamic behavior such as boundary crisis and internal crisis, which will cause the chaotic attractor to disappear and become larger.