Abstract

This paper is the sixth in a series of papers devoted to two-dimensional homogeneous cubic systems. It considers a case where a homogeneous vectorial polynomial in the right-hand part of the system does not have a common multiplier. A set of such systems is divided into classes of linear equivalence; in each of them, the simplest system is a third-order normal form which is separated on the basis of properly introduced principles. Such a form is defined by the matrix of its right-hand part coefficients, which is called the canonical form (CF). Each CF has its own arrangement of non-zero elements, their specific normalization and a canonical set of permissible values for the unnormalized elements, which relates the CF to the selected equivalence class. In addition to the classification, each CF is provided with: a) the conditions on the coefficients of the initial system, b) non-singular linear substitutions that reduce the right-hand side of the system under these conditions to the selected CF, c) obtained values of CF’s unnormalized elements. The proposed classification was primarily created to obtain all possible structures of generalized normal forms for the systems with a CF in the unperturbed part. This paper presents another application of the resulting classification related to finding phase portraits in the Poincare circle for the CF.

中文翻译：

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二维齐次三次系统：分类和正规形式—VI

摘要

本文是有关二维齐次三次系统的系列文章中的第六篇。它考虑了系统右侧的齐次矢量多项式没有公共乘数的情况。一组这样的系统分为线性等价类。在它们每个中，最简单的系统是三阶范式，它是根据适当引入的原理分开的。这种形式由其右边部分系数的矩阵定义，这称为规范形式（CF）。每个CF都有其自己的非零元素安排，它们的特定归一化以及未归一化元素的一组规范化的允许值，这将CF与选定的等效类相关联。除分类外，每个CF还提供：a）初始系统系数的条件，b）在这些条件下将系统的右侧缩小为所选CF的非奇异线性替换，c）获得CF的未归一化元素的值。最初创建建议的分类是为了获得在不受干扰的部分具有CF的系统的所有可能的广义范式的结构。本文介绍了所得分类的另一种应用，该分类与在CF的Poincare圆中查找相像有关。最初创建建议的分类是为了获得在不受干扰的部分具有CF的系统的所有可能的广义范式的结构。本文介绍了所得分类的另一种应用，该分类与在CF的Poincare圆中查找相像有关。最初创建建议的分类是为了获得在不受干扰的部分具有CF的系统的所有可能的广义范式的结构。本文介绍了所得分类的另一种应用，该分类与在CF的Poincare圆中查找相像有关。