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 the case where homogeneous polynomial vector on the right-hand part of the system has a square common factor with complex 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 and is 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 relates CF to the selected class of equivalence. In addition, for each CF, (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, and (c) obtained values of CF’s non-normalized elements are given. Refs 9.