Normal forms for coupled Takens-Bogdanov systems
The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and is best described by giving a Stanley decomposition of that module. Groebner basis methods are used to determine the Stanley decomposition of the ring of invariants, that arise in normal forms for systems with nilpotent linear part consisting of repeated 2 x 2 Jordan blocks. Then an efficient algorithm developed by Murdock, is used to produce a Stanley decomposition of the module of the equivariants from the Stanley decomposition of the ring of invariants. Also included is a discussion of the phenomenon of asymptotic unfolding and is used to find the unfolding of single Takens-Bogdanov systems.