Normal forms of Hopf-zero singularity

作者：Gazor Majid^*; Mokhtari Fahimeh

来源：Nonlinearity, 2015, 28(2): 311-330.

DOI：10.1088/0951-7715/28/2/311

• Normal form
• Hopf-zero singularity
• sl(2)-representation
• Conservative and nonconservative decomposition

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 computations of Bogdanov-Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic nonzero 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 Rossler and Kuramoto-Sivashinsky equations to demonstrate the applicability of our results.

• 出版日期2015-2