Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Canonical Forms of Two-dimensional Homogeneous Cubic Systems with a Common Square Factor

Author(s):

Vladimir V. Basov

Universitetsky prospekt, 28,
198504, Peterhof, St. Petersburg, Russia,
Saint-Petersburg State University,
The Faculty of Mathematics and Mechanics,
Differential Equations Department

vlvlbasov@rambler.ru

Aleksander Chermnykh

Universitetsky prospekt, 28, 198504, Peterhof, St. Petersburg, Russia,
Saint-Petersburg State University,
The Faculty of Mathematics and Mechanics, Differential Equations Department

achermnykh@yandex.ru

Abstract:

Real two-dimensional homogeneous cubic systems of ODE for which polynomials in right-hand parts have a common square factor are considered. In accordance with properly introduced ordering principles the set of these systems is divided into linear equivalence classes such that in each class the structurally simplest system is distinguished --- the normal form of the third order. For the coefficient matrix of the normal form right-hand side (the canonical form -- CF) the canonical set of values ensuring the system belonging to a selected class is specified. In addition, for each CF a) conditions on the coefficients of the original system, b) linear substitutions reducing the right-hand part of the system under these conditionsto to the chosen CF, and c) obtained values of CF coefficients are given. In applications the programs written with using Maple software and allowing to obtain the majority of practical results are presented.

Keywords

• canonical form
• homogeneous cubic system
• normal form

References:

1. Basov V. V. Two-Dimensional Homogeneous Cubic Systems: Classification and Normal Forms. I, Vestnik St. Petersburg University. Mathematics, 49(2), 99-110 (2016). Available at http://link.springer.com/article/10.3103/S1063454116020023
2. Basov V. V. Two-Dimensional Homogeneous Cubic Systems: Classification and Normal Forms. II, Vestnik St. Petersburg University. Mathematics, 49(3), 204-218 (2016)

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