Unique normal forms: The nilpotent Hamiltonian case

The problem we consider in this paper, namely the characterization of unique normal forms, derives from bifurcation theory. In a typical situation, one has an unperturbed vector field near a degenerate equilibrium. In order to study the effect of generic perturbations, suitable coordinates are introduced and the simplified system is said to be in normal form. A well known case is the Hopf bifurcation where a pair of conjugate eigenvalues cross the imaginary axis. Using averaging, one can in this case compute the periodic orbit that splits off from the equilib~um, A second familiar example is the Takens-Bogdanov bifurcation, where two eigenvalues are zero. In all these situations it is necessary to make certain choices, both in the definition of what constitutes a “normal form” and in the computation of the transformation. The idea here is to use this second kind of choice to sharpen the very definition of normal form. While in standard treatments only the linear part of the field plays a role, we show that one can effectively use its non-linear part to remove additional terms, and thus simplify the problem further. The underlying philosophy in this type of game is that