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We investigate the breakdown of normal hyperbolicity of a manifold ofequilibria of a flow. In contrast to classical bifurcation theory we assume theabsence of any flow-invariant foliation at the singularity transverse to themanifold of equilibria. We call this setting bifurcation without parameters. Inthe present paper we provide a description of general systems with a manifoldof equilibria of codimension one as a first step towards a classification ofbifurcations without parameters. This is done by relating the problem tosingularity theory of maps.

Dynamics near manifolds of equilibria of codimension one and bifurcation without parameters