Normal forms of Hopf-zero singularity

Abstract

The Lie algebra generated by Hopf-zero classical normal forms is decomposed into two versal Lie subalgebras. Some dynamical properties for each subalgebra are described; one is the set of all volume-preserving conservative systems while the other is the maximal Lie algebra of nonconservative systems. This introduces a unique conservativenonconservative decomposition for the normal form systems. There exists a Lie-subalgebra that is Lie-isomorphic to a large family of vector fields with Bogdanov-Takens singularity. This gives rise to a conclusion that the local dynamics of formal Hopf-zero singularities is well-understood by the study of Bogdanov-Takens singularities. Despite this, the normal form computation of Bogdanov-Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic non-zero condition, complete results on the simplest Hopf-zero normal forms are obtained in terms of the conservative-nonconservative decomposition. Some practical formulas are derived and the results implemented using Maple. The method has been applied on the Rössler and Kuramoto-Sivashinsky equations to demonstrate the applicability of our results.

On the representations andZ2-equivariant normal form for solenoidal Hopf-zero singularities

“…. This class of vector fields was derived by sl 2 -decomposition of the classical normal form of Hopf-zero bifurcation [17]. And indeed, the above system could be formulated using the Lie algebraic structure as follows.…”

Normal forms for Hopf-Zero singularities with nonconservative nonlinear part

“…and E is considered as a differential operator acting on functions, see [9, p. 2] and [12]. The structure constants follow from the fact that…”

“…Our normal form computation is to use a transformation group that preserves the Eulerian structure and it would only simplify the function f . However, our results in [12] work with the bigger group, that is, the set of all near-identity transformations and may create conservative terms into the normal form system. Indeed, the quasi-Eulerian vector field E is defined such that its generated family would form a Lie algebra.…”

“…Although the Hamiltonian vector fields (symplectic structures) require an even dimensionality, the decomposition idea still can work. Indeed we instead work with conservative and nonconservative family of vector fields, see [12]. In this paper we are concerned with the study of nonconservative family.…”

“…The reader should note that this family of vector fields are indeed the classical (first level) normal form of a very much larger family of vector fields. This, however, is independent from [12] and basically refers to an essentially separate problem. Our normal form computation is to use a transformation group that preserves the Eulerian structure and it would only simplify the function f .…”