Glasnik Matematicki, Vol. 50, No. 1 (2015), 163-182.

ON QUADRATIZATIONS OF HOMOGENEOUS POLYNOMIAL SYSTEMS OF ODES

Matej Mencinger

Faculty of Civil Engineering, University of Maribor, 2 000 Maribor, and, IMFM, 1 000 Ljubljana, Slovenia
e-mail: matej.mencinger@um.si

Abstract.   The quadratizations of a (homogeneous nonquadratic) nonlinear polynomial system of ODEs introduced by Myung and Sagle in [17] is considered. The 1-1 correspondence between homogeneous quadratic systems of ODEs and nonassociative algebras is used to prove a special structure of the algebra corresponding to a general homogeneous quadratic systems being a quadratization. Every homogeneous solution-preserving map (corresponding to a quadratization) determines the so called essential set which turns out to be crucial for preserving the (in)stability of the origin from homogeneous nonquadratic systems to their quadratizations and vice versa. In particular the quadratizations of homogeneous systems x' =fα(x) (of order α>2) and cubic planar systems are considered. In the main result we prove that for quadratizations of cubic planar systems the (in)stability of the origin is preserved from the original system x→' =fα(x→) , α>2 to the quadratization (regarding the essential set of the corresponding solution-preserving map) and vice versa.

2010 Mathematics Subject Classification.   34A34, 34D20, 13P99.

Key words and phrases.   Homogeneous system, cubic system, quadratic system, quadratization, commutative (nonassociative) algebra, stability, critical point.

Full text (PDF) (free access)

DOI: 10.3336/gm.50.1.09

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