A method is presented for the direct determination and stability analysis of periodic states of adsorption cycles. The method is shown to be more versatile than the traditional method of successive substitution and computationally more efficient for cycles which converge slowly to the periodic state. Examples of unconstrained and constrained cycles are given based on the pressure swing adsorption (PSA) model of Ritter and Yang. For the constrained case, the first and only periodic state found satisfies a design specification.