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SUMMARYNon-linear static buckling of simple systems associated with typical discrete critical points is comprehensively presented using elementary Catastrophe Theory. Attention is focused on the Fold and Cusp Catastrophe, all local properties of which are assessed in detail. Hence, in dealing with stability problems of potential systems there is no need to seek any of these properties since all of these are known a priori. Then, one has only to classify, after reduction, the total potential energy of a system into one of the universal unfoldings of the above types of catastrophe. Two illustrative numerical examples show the methodology of the proposed technique.