Two-dimensional homogeneous cubic systems: Classification and normal forms. I

This work is the first in a series of papers devoted to classifying of two-dimensional homogeneous cubic systems based on partitioning into classes of linear equivalence. Principles have been developed that are capable of constructively distinguishing the structure of a simplest system in each class and a canonical set that defines the admissible values that can be assumed by its coefficients. The polynomial vector in the right-hand part of this system identified with a 2 × 4 matrix is called the canonical form (CF) and the system itself is called the normal cubic form. One of the main objectives of this series of papers is to maximally simplify the reduction of a system with a homogeneous cubic polynomial in the unperturbed part to the various structures of a generalized normal form (GNF). Generalized normal form refers to a system in which the perturbed part has the simplest form in some sense. The constructive implementation of the normalization process depends on the ability to explicitly specify the conditions of compatibility and possible solutions of the so-called bonding system, which is understood to be a countable set of linear algebraic equations that specify the normalizing transformations of the perturbed system. The above principles are based on the idea of the maximum possible simplification of the bonding system. This will allow one to first reduce the initial perturbed system by an invertible linear substitution of variables to a system with some CF in the unperturbed part, then reduce the resulting system, which is optimal for normalization, by almost identical substitutions to various structures of the GNF. In this paper, the following tasks are carried out: (1) the general problem is set, close problems are formulated, and the available results are described; (2) a bonding system is derived that is capable of determining the equivalence of any two perturbed systems with the same homogeneous cubic part, the possibilities of its simplification are discussed, the GNF is defined, and the method of resonant equations is given allowing one to constructively obtain all its structures; (3) special forms of recording homogeneous cubic systems in the presence of a common homogeneous factor in their right-hand parts with a degree of 1–3 are introduced, and the linear equivalence of these systems, as well as of systems without a common factor is studied, and key linear invariants are offered. © 2016, Allerton Press, Inc.