Solution spaces for adsorption cycles are mapped out using a continuation method. As examples, we consider adsorption cycles that exhibit multiplicity. Within a region of multiplicity, a cycle can settle into more than one periodic state for a single operating specification or into a periodic state that repeats itself not every cycle but every two cycles. The stabilities of the periodic states are assessed to identify stable and unstable states and to determine the manner in which the cycles lose stability. Application of the method is illustrated for a temperature swing adsorption cycle (Davis et al., 1988, Ind. Engng Chem. Res. 27,1229-1235; LeVan, 1990, Ind. Engng Chem. Res. 29, 625-631) and a pressure swing adsorption cycle (Suh and Wankat, 1989, A.I.Ch.E. J. 35, 523-526).