A single-degree-of-freedom quarter suspension system is considered. According to the theory of Floquet, we can get the Floquet multiplier which is employed to determine the stabled of period trajectory and the Lyapunov exponent which is employed to determine the nature of chaos motion. We reveal a new hysteresis bifurcation: hysteresis loop is composed of a stable period trajectory and an unstable period trajectory, the stable period trajectory which has a complete priod-dobubling sequence. The priod-dobubling sequence has appeared a chaos trajectory on the boundary of the hysteresis loop, so there will occur jumping phenomena between the stable period trajectory and the chaos trajectory on the boundary of the hysteresis loop as the change of parameters. In addition, if the chaos trajectory formed by priod-dobubling sequence meets with unstable trajectory on the boundary of the hysteresis loop, the chaos trajectory will occur boundary crisis and disappeared suddenly, then jumps to the other stable trajectory. According to the method of linear augmentation control, the multistability is controlled which insides of hysteresis loop, including the high amplitude 3-period trajectory has controlled to low amplitude 1-period trajectory and 1-period motion has controlled to chaos motion. The results of the research can provide theoretical reference for the dynamics design of vehicle suspension.