title: Two-dimensional homogeneous cubic systems: classification and normal forms — I
metadata.name: dc.contributor.author
metadata.value: Basov, Vladimir V.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2016-08-01T10:29:20Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2016-08-01T10:29:20Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2016-06
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/2454
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: This work is the ﬁrst in a series of papers devoted to the classiﬁcation of two-dimensional homogeneous cubic systems based on the partition into classes of linear equivalence. Principles are developed to constructively distinguish the structure of the simplest system in each class and canonical set that deﬁnes the permissible values that can take its coeﬃcients. The polynomial vector in the right part of such system identiﬁed with (2×4)-matrix is called the canonical form (CF) and the system itself — the normal cubic form.
  One of the main objectives of the series is to simplify the reduction of a system with a homogeneous cubic polynomial in the unperturbed part to the various structures of the generalized normal form (GNF). Under GNF we mean a system, perturbed part of which is in some sense the simplest form. Constructive implementation of the normalization process depends on the ability to explicitly specify the conditions of compatibility and possible solutions of so-called connective system which is understood as a countable set of linear algebraic equations systems that determine the normalizing transformation of the perturbed system. The principles mentioned above are based on the idea of the greatest possible simpliﬁcation of the connective system. This will allow to reduce the initial perturbed system by an invertible linear substitution to the system with some CF in the unperturbed part, and then reduce the resulting system, optimal for normalization, by almost identical substitutions to various structures of the GNF.
  In this paper: 1) the general problem is set, the other problems close to it are formulated with the description of the existing results; 2) the connective system is derived, which determines the equivalence of any two perturbed systems with the same homogeneous cubic part, the possibilities of its simpliﬁcation are discussed, also the GNF is deﬁned and the method of resonant equations which allows to constructively obtain all of its structures is given; 3) special recording forms of homogeneous cubic systems in the presence of the common homogeneous factor in their right-hand parts having a power from one to three are introduced; the linear equivalence of such systems as well as systems without a common factor is studied; the key linear invariants are oﬀered. Refs 20.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of St Petersburg University. Series 1. Mathematics. Mechanics. Astronomy;Vol. 3 (61); Issue 2
metadata.lang: -
metadata.name: dc.subject
metadata.value: homogeneous cubic system
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: normal form
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: canonical form
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Two-dimensional homogeneous cubic systems: classification and normal forms — I
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: Басов.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/2454/1/%d0%91%d0%b0%d1%81%d0%be%d0%b2.pdf
resources.format: Adobe PDF
resources.size: 318,5 kB