Volume-preserving normal forms of Hopf-Zero singularity

Details Volume-preserving normal forms of Hopf-Zero singularity Volume-preserving normal forms of Hopf-Zero singularity

2012-03-08

arxiv.org/abs/1203.1807v1

Mathematics

Dynamical Systems

Scientific paper

A practical method is described for computing the generators of the algebra of first integrals that works well for a large family of Hopf-Zero singularity systems. A Lie algebra description is presented for all volume-preserving classical normal form of Hopf-Zero singularity systems. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a non-zero condition on their quadratic part are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any non-degenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are computed. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and implemented on appropriately modified R\"{o}ssler and generalized Kuramoto-Sivashinsky equations to demonstrate the applicability of our theoretical results.

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