An improved theory of asymptotic unfoldings

Abstract

An asymptotic unfolding of a dynamical system near a rest point is a system with additional parameters, such that every oneparameter deformation of the original system can be embedded in the unfolding preserving all properties that can be detected by asymptotic methods. Asymptotic unfoldings are computed using normal (and hypernormal) form methods. We present a simplified and improved method of computing such unfoldings that can be used in any normal form style.

Box products in nilpotent normal form theory: The factoring method

“…This paper is confined to the first part of such a program. The second (unfolding) is addressed in [23], and the third part (the blow-ups) should probably be addressed by specialists in the techniques of Takens, Bogdanov, A.D. Bruno [6,7], and F. Dumortier [13].…”

“…The simplified normal form occurred first in [15] in a few nilpotent examples. It was generalized to matrices that are not necessarily nilpotent in [22, §4.6], and used in [21] and [23]; for non-nilpotent cases the simplified normal form retains its (useful) equivariance with respect to the semisimple part of the linear term. (The earlier paper [20], and Section 6.4 of [22], are obsolete and are replaced by [23].…”

Normal forms for Hopf-Zero singularities with nonconservative nonlinear part

“…(1.1), see also [10,16,21]. Define extended from nonparametric to the parametric cases, see e.g., [15,16].…”

Bifurcation Control and Universal Unfolding for Hopf-Zero Singularities with Leading Solenoidal Terms

“…Murdock [35,36] defines two systems as n-equivalent when they share all their n-jet (n-degree Taylor expansion) determined properties. The n-asymptotic unfolding is defined based on the n-equivalence relation and it seems the most natural way of defining a versal unfolding amenable to computation and normal form analysis; e.g., see [39,Theorem 1]. We slightly modify versal asymptotic unfolding and call it versal asymptotic unfolding normal form, that is, a versal asymptotic unfolding for our simplest (orbital) normal form system.…”

“…This section is devoted to treat Hopf-zero singularities (whose the first few dominant terms are solenoidal) with any possible additional nonlinear-degeneracies. Here, the n-equivalence relation (n-jet determined) is used for introducing universal asymptotic unfolding normal form, whose original ideas are due to [35,36,39] and is amenable to finite normal form computations. Consider the parametric differential equatioṅ…”