Ok. Looks interesting but unfortunately I don't have time right now to work with it. May I suggest taking a look at "An Introductioon to Catastrophe Theory" by Saunders and try and adapt your equation to the canonical version of the swallowtail:
I did notice when you put yours over a common denominator, the numerator is a quartic but it includes a cubic term which the canonical swallowtail does not include. Not sure how that would effect the bifurcation set. So the general procedure is to then take the derivative of the RHS, then set the RHS and it's derivative to zero and then eliminate x from these two expressions. This then gives an implicit equation in u, v, and w. That surface is the swallowtail bifurcation set. However, your equation has more than three parameters. Not sure about this also but I would start by trying to fit your equation to the canonical version even if I have to simplify it or constrain it.