TY - JOUR

T1 - Two-dimensional homogeneous cubic systems: Classification and normal forms II

AU - Basov, V. V.

N1 - Basov, V.V. Two-dimensional homogeneous cubic systems: Classification and normal forms II. Vestnik St.Petersb. Univ.Math. 49, 204–218 (2016). https://doi.org/10.3103/S1063454116030031

PY - 2016

Y1 - 2016

N2 - This work is the second in a series of papers concerning two-dimensional homogeneous cubic systems. In the first paper of the series, structural principles were developed to introduce a total order on the set of structural forms, i.e., vector polynomials with a fixed number of zero coefficients that are right-hand sides of two-dimensional homogeneous cubic systems of ODEs. Among them,structural forms normalized on the basis of normalization principles and canonical forms (CFs) that are linearly nonequivalent to each other and are the simplest in their class were sequentially distinguished. In this paper, for above-mentioned systems with proportional right-hand side components, all CFs with their canonical sets of permissible values are distinguished. For each CF, (a) conditions on the coefficients of the original system, (b) linear substitutions that reduce the right-hand side of a system under these conditions to the chosen CF, and (c) the resulting values of the CF’s coefficients are given.

AB - This work is the second in a series of papers concerning two-dimensional homogeneous cubic systems. In the first paper of the series, structural principles were developed to introduce a total order on the set of structural forms, i.e., vector polynomials with a fixed number of zero coefficients that are right-hand sides of two-dimensional homogeneous cubic systems of ODEs. Among them,structural forms normalized on the basis of normalization principles and canonical forms (CFs) that are linearly nonequivalent to each other and are the simplest in their class were sequentially distinguished. In this paper, for above-mentioned systems with proportional right-hand side components, all CFs with their canonical sets of permissible values are distinguished. For each CF, (a) conditions on the coefficients of the original system, (b) linear substitutions that reduce the right-hand side of a system under these conditions to the chosen CF, and (c) the resulting values of the CF’s coefficients are given.

KW - homogeneous cubic system

KW - normal form

KW - canonical form

UR - https://link.springer.com/article/10.3103/S1063454116030031

M3 - Article

VL - 49

SP - 204

EP - 218

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -