Two-dimensional homogeneous cubic systems: classification and normal forms — I

Abstract

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.