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

Abstract

This article is the third in a series of works devoted to the two-dimensional cubic homogeneous systems. It is considered a case when a homogeneous polynomial vector in the right-hand part of the system has a square common factor with real zeros. The set of such systems is divided into classes of linear equivalence, in each of them on the basis of properly introduced structural and normalization principles the simplest system is distinguished the normal form of the third order. In fact, the normal form is defined by the coefficient matrix of the right-hand part, 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 non-normalized elements, which guarantees CF’s belonging to the selected class of equivalence. In addition, for each CF are given: a) the conditions on the coefficients of the initial system, b) non-singular linear substitution reducing the right-hand part of the system under these conditions to the selected CF, c) obtained values of CF’s non-normalized elements. Refs 6.