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arXiv · 1210.4467

Normal Forms of Hopf-zero singularity

Majid Gazor·Fahimeh Mokhtari
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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 Rössler and Kuramoto--Sivashinsky equations to demonstrate the applicability of our results.

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BibTeXRIS

Majid Gazor, Fahimeh Mokhtari. 2014-10-31. Normal Forms of Hopf-zero singularity. https://doi.org/10.1088/0951-7715%2F28%2F2%2F311

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