TY - JOUR

T1 - Two-Dimensional Homogeneous Cubic Systems

T2 - Classification and Normal Forms—VI

AU - Basov, V. V.

AU - Chermnykh, A. S.

N1 - Publisher Copyright: © 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - This paper is the sixth in a series of papers devoted to two-dimensional homogeneous cubic systems. It considers a case where a homogeneous vectorial polynomial in the right-hand part of the system does not have a common multiplier. A set of such systems is divided into classes of linear equivalence; in each of them, the simplest system is a third-order normal form which is separated on the basis of properly introduced principles. Such a form is defined by the matrix of its right-hand part coefficients, which is called the canonical form (CF). Each CF has its own arrangement of non-zero elements, their specific normalization and a canonical set of permissible values for the unnormalized elements, which relates the CF to the selected equivalence class. In addition to the classification, each CF is provided with: a) the conditions on the coefficients of the initial system, b) non-singular linear substitutions that reduce the right-hand side of the system under these conditions to the selected CF, c) obtained values of CF's unnormalized elements. The proposed classification was primarily created to obtain all possible structures of generalized normal forms for the systems with a CF in the unperturbed part. This paper presents another application of the resulting classification related to finding phase portraits in the Poincare circle for the CF.

AB - This paper is the sixth in a series of papers devoted to two-dimensional homogeneous cubic systems. It considers a case where a homogeneous vectorial polynomial in the right-hand part of the system does not have a common multiplier. A set of such systems is divided into classes of linear equivalence; in each of them, the simplest system is a third-order normal form which is separated on the basis of properly introduced principles. Such a form is defined by the matrix of its right-hand part coefficients, which is called the canonical form (CF). Each CF has its own arrangement of non-zero elements, their specific normalization and a canonical set of permissible values for the unnormalized elements, which relates the CF to the selected equivalence class. In addition to the classification, each CF is provided with: a) the conditions on the coefficients of the initial system, b) non-singular linear substitutions that reduce the right-hand side of the system under these conditions to the selected CF, c) obtained values of CF's unnormalized elements. The proposed classification was primarily created to obtain all possible structures of generalized normal forms for the systems with a CF in the unperturbed part. This paper presents another application of the resulting classification related to finding phase portraits in the Poincare circle for the CF.

KW - canonical form

KW - homogeneous cubic system

KW - normal form

UR - http://www.scopus.com/inward/record.url?scp=85090028276&partnerID=8YFLogxK

U2 - 10.1134/S1063454120030048

DO - 10.1134/S1063454120030048

M3 - Article

AN - SCOPUS:85090028276

VL - 53

SP - 248

EP - 260

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -