The onset of the expansion phase of substorms is currently unpredictable, that unpredictability being related to the nonlinear nature of the solar wind - magnetosphere interaction. Using the mathematic of catastrophe theory it is possible to describe the substorm phenomenon in terms of a fundamental singularity of the solar wind - magnetosphere interaction. The apparently random nature of substorm onset can then be explained as the result of an unfolding of the singularity of the cusp catastrophe which is in principle predictable. Substorm expansions are triggered when two features of the solar wind - magnetosphere interaction are critically functionally related. ¿American Geophysical Union 1991