Two-Dimensional Homogeneous Cubic Systems: Classification and Normal Forms II

This work is the second in a series of papers concerning two-dimensional homogeneous cubic systems. In the first paper of the series, structural principles were developed to introduce a total order on the set of structural forms, i.e., vector polynomials with a fixed number of zero coefficients that are right-hand sides of two-dimensional homogeneous cubic systems of ODEs. Among them, structural forms normalized on the basis of normalization principles and canonical forms (CFs) that are linearly nonequivalent to each other and are the simplest in their class were sequentially distinguished. In this paper, for above-mentioned systems with proportional right-hand side components, all CFs with their canonical sets of permissible values are distinguished. For each CF, (a) conditions on the coefficients of the original system, (b) linear substitutions that reduce the right-hand side of a system under these conditions to the chosen CF, and (c) the resulting values of the CF's coefficients are given.