Two-dimensional homogeneous cubic systems: Classification and normal forms: IV

Abstract

This article is the fourth in a series of works devoted to two-dimensional cubic homogeneous systems. It considers a case when a homogeneous polynomial vector in the right-hand part of the system has a quadratic common factor with complex zeros. A set of such systems is divided into classes of linear equivalence, wherein the simplest system is distinguished on the basis of properly introduced structural and normalization principles, being, thus, the third-order normal form. In fact, such a form is defined by a matrix of its right-hand part coefficients, which is called the canonical form (CF). Each CF has its own arrangement of nonzero elements, their specific normalization and canonical set of permissible values for the nonnormalized elements, which relates CF to a selected class of equivalence. In addition, each CF is characterized by: (1) conditions imposed on the coefficients of the initial system, (2) nonsingular linear substitutions that transform the right-hand part of the system under these conditions into a selected CF, and (3) obtained values of CF’s nonnormalized elements. Refs 9.