Parametric normal form classification for Eulerian and rotational non-resonant double Hopf singularities
Abstract
In this paper we provide novel results on the infinite level normal form and orbital normal form classifications of nonlinear Eulerian and rotational vector fields with two pairs of non-resonant imaginary modes. We use the method of multiple Lie brackets and its extension along with time rescaling for orbital normal form classification. Furthermore, we apply two reduction techniques. The first is to use the radical Lie ideal of rotational vector fields and its corresponding quotient Lie algebra. The second technique is to employ a Schur complement block matrix type in Gaussian elimination and analysis of block matrices. The infinite level parametric normal form classification are also presented. The latter is also viewed as a normal form result for multiple-input controlled systems with non-resonant double Hopf singularity. We also discuss nonlinear symmetry transformations associated with the nonlinear symmetry group of the simplest normal forms. Symbolic normal form transformation generators are derived for computer algebra implementation. Further, the results are efficiently implemented and verified using Maple for all three types of normal form computations up to arbitrary degree, where they can also include both small bifurcation parameters and arbitrary symbolic constant coefficients.
Cite
@article{arxiv.1812.11528,
title = {Parametric normal form classification for Eulerian and rotational non-resonant double Hopf singularities},
author = {Majid Gazor and Ahmad Shoghi},
journal= {arXiv preprint arXiv:1812.11528},
year = {2019}
}
Comments
Some proofs are omitted on this ArXiv version for briefness and full draft paper is available upon request from authors. Some materials from previous arXiv version of the paper have been excluded from this version of the draft. They will be improved and extended to appear elsewhere