Local diffeomorphisms in the bifurcational manifestations of the umbilic catastrophes

Typical imperfection-sensitivity plots for semi-symmetric points of bifurcation are presented in three-dimensional control space, giving rise to the universally-unfolded forms of the hyperbolic and elliptic umbilic catastrophes. The structurally-stable distorting of these surfaces is then observed with the addition of a fourth control parameter, which separates the two contributing bifurcations of the perfect system. A further consequence of structural stability is then explored. Simple transformations are presented that allow all the general forms of imperfection-sensitivity to be interpreted for specific problems, here a simple guyed cantilever model under a number of different initial conditions. This is only made possible by local diffeomorphisms which are known to be present, and which result directly from the underlying topological classification of catastrophe theory.