I have a dynamical system restricted to it's (1d-) Center Manifold. All coefficients of the 'pure' variable up to a certain order are zero. There are not enough parameters to fully unfold the bifurcation, e.g.:

p1+p2*x+x^4, if there was one more paramter it would be a swallowtail, but what would this one be? or similarly

p1(q1,q2)+p2(q1,q2)*x+p3(q1,q2)*x^2+x^4 (basically there are only two independent parameters)