Normal Form in Filtered Lie Algebra Representations

An improved theory of asymptotic unfoldings

“…Basic references on hypernormal forms include [3] and [1]. For an introduction and a critical survey of the literature on hypernormal forms before 2003, see [7, §4.10], and for some recent developments, see [8][9][10][11].…”

Spectral sequences and parametric normal forms

“…Arnold [3,4] was the first to apply this method on singularity and normal form theory but it were not later used by others. Recently Sanders [25,26] further elaborated the method and evidently encouraged a few researchers in using this method on normal form theory [9,21]. Sanders [25,26] and Murdock [21] mainly focus on normal forms of non-parametric vector fields while Benderesky and Churchill [9] applied the method on matrix normal forms.…”

“…We provide an example of detailed calculations of hypernormal forms using the notations of the spectral sequences in section 4. Spectral sequences have been introduced for normal form calculations in [9,21,25,26] but even in these papers, the spectral sequence method is usually kept in the background as a guide to the calculations, while the calculations themselves are done in the ordinary way. Here we show what it would look like to employ the notations of the spectral sequences in full.…”

Infinite Order Parametric Normal Form of Hopf Singularity

“…Interestingly, some of these results sometimes naively seem unrelated but they end up being quite helpful. For example, Sanders [2003Sanders [ , 2005 has recently paid attention to applications of cohomology theory and spectral sequences in the computation of normal forms. Kokubu et al [1996] raised the notion of multiple Lie bracket method and quasi homogeneous normal form theory, see also [Algaba et al, 2003;Ashkenazi & Chow, 1988;Chen & Della Dora, 2000;Chen et al, 2000;Peng & Wang, 2004;Wang et al, 2000].…”