Normal Forms of Hopf-zero singularity

TL;DR

This paper decomposes the Lie algebra of Hopf-zero normal forms into conservative and nonconservative parts, providing a comprehensive understanding of their local dynamics and practical formulas for computation, with applications to well-known equations.

Contribution

It introduces a unique conservative--nonconservative decomposition of Hopf-zero normal forms and links their dynamics to Bogdanov--Takens singularities, advancing the theoretical framework.

Findings

Decomposition of Lie algebra into two subalgebras.

Complete results on simplest Hopf-zero normal forms under quadratic condition.

Application of formulas to R"ossler and Kuramoto--Sivashinsky equations.

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 conservative--nonconservative 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 R\"ossler and Kuramoto--Sivashinsky equations to demonstrate the applicability of our results.