Asymptotic Unfoldings of Dynamical Systems by Normalizing beyond the Normal Form

Systematic methods are given for simplification of the normal form of a perturbed dynamical system near a rest point. It then becomes easy to write down an unfolding of the dynamical system which can be rigorously justified as a universal unfolding with respect to asymptotic equivalence. These unfoldings always have finite codimension. They are both weaker and stronger than topological unfoldings: they may fail to be universal unfoldings in the topological sense because of behavior which escapes detection by asymptotic methods, and they also may distinguish asymptotic behaviors that are topologically equivalent. The method is illustrated by computing an unfolding of codimension 14, as well as some standard examples of codimension 2 and 3.