ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

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

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

### Author(s):

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

vlvlbasov@rambler.ru

Aleksandr Chermnykh

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

achermnykh@yandex.ru

### Abstract:

Real two-dimensional homogeneous cubic systems of ODE, polynomials in the right-hand part of which have a common linear factor are considered. On the basis of properly introduced ordering principles, the set of these systems is divided into classes of linear equivalence, in each of which the structurally simplest system --- the third order normal form --- is distinguished. For the coefficient matrix of its right-hand side, called the canonical form (CF), the canonical set of values is specified, which guarantees the affiliation of the system to the selected class. In addition, for each CF a) conditions on the coefficients of the original system, b) linear substitutions that reduce the right-hand part of the system under these conditions to the chosen CF, c) obtained values of CF coefficients are given. In existing applications programs written using the Maple software are presented, which allowed us to obtain the majority of practical results. Refs 4.

### Keywords

• каноническая форма
• нормальная форма
• однородная кубическая система

### 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) (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) (https://link.springer.com/article/10.3103/S1063454116030031)
3. Basov V. V., Chermnykh A. S., Two-Dimensional Homogeneous Cubic Systems: Classification and Normal Forms - III, Vestnik St. Petersburg University. Mathematics 50 (2), 97-110 (2017) (https://link.springer.com/article/10.3103/S1063454117020029)
4. Basov V. V., Chermnykh A. S., Two-Dimensional Homogeneous Cubic Systems: Classification and Normal Forms - IV, Vestnik St. Petersburg University. Mathematics 50 (3), 217-234 (2017) (https://link.springer.com/article/10.3103/S1063454117030049)