English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we provide novel results on the infinite level normal form andorbital normal form classifications of nonlinear Eulerian and rotational vectorfields with two pairs of non-resonant imaginary modes. We use the method ofmultiple Lie brackets and its extension along with time rescaling for orbitalnormal form classification. Furthermore, we apply two reduction techniques. Thefirst is to use the radical Lie ideal of rotational vector fields and itscorresponding quotient Lie algebra. The second technique is to employ a Schurcomplement block matrix type in Gaussian elimination and analysis of blockmatrices. The infinite level parametric normal form classification are alsopresented. The latter is also viewed as a normal form result for multiple-inputcontrolled systems with non-resonant double Hopf singularity. We also discussnonlinear symmetry transformations associated with the nonlinear symmetry groupof the simplest normal forms. Symbolic normal form transformation generatorsare derived for computer algebra implementation. Further, the results areefficiently implemented and verified using Maple for all three types of normalform computations up to arbitrary degree, where they can also include bothsmall bifurcation parameters and arbitrary symbolic constant coefficients.

Parametric normal form classification for Eulerian and rotational non-resonant double Hopf singularities