An undamped model for the response of one DOF(degreeoffreedom) systems having twosided amplitude constraints was proposed. Firstly, the Poincaré map of the periodn motions was set up for the system by using fixed phase plane as Poincaré section, and the stability of periodic motions was also discussed. Then a numerical simulation was carried out, and it was shown that our conclusions were valid. Moreover, the phenomenon of multisolutions’ coexistence in this system was found through the numerical simulation. Finally, a means of pullback integral is introduced into the impact process based on the cellmapping method, from which the coexistent attractors of the symmetric doubleimpact period1 motions were derived, and it was found that the variation of domains of attraction was related to the leap phenomenon after comparing with Poincaré map figures.