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Multistability of a dynamic system consists of the ability to express multiple stable states and to switch between these states in response to some external input. From a viewpoint of dynamical system theory, the multistability and bifurcation phenomena of supersonic inlet start/unstart under the assumption of inviscid flow is analytically described. Bifurcation analysis shows the existence of multistability will result in diverse hysteresis properties in the transitions of the multiple flow patterns, and the transition routes will be considerably complex. Guided by the general principles of multistability systems, the loops-coupled hysteresis effects of inlet start/unstart are reported and analyzed. The loops-coupled hysteresis, associated with the interaction among the multiple inlet flow patterns, indicates strong route dependence of multiple flow pattern transitions. The complexity of dynamics behaviors involving loops-coupled hysteresis and complex routes originates from the memory effect of initial states and the existence of multisolution domain. The control of bifurcation system thus depends not only on the instantaneous value of the input but also on the history of its operation. A better understanding of multistability and complex routes of inlet start/unstart may be useful as a guide in thinking about the bifurcation mechanism, designing the control routes, and actively controlling the bifurcation behaviors of inlet start/unstart.