Abstract

In this paper, the study of the structural arrangement of generalized normal forms (GNFs) is continued. A planar system that is real-analytic at the origin of coordinates is considered. Its unperturbed part forms a first degree quasi-homogeneous polynomial (\(\alpha x_{1}^{2}\) + x₂, x₁x₂) with weight (1, 2), in which parameter α ∈ (–1/2, 0) ∪ (0, 1/2]. For the given values of α, this polynomial is a generatrix in the canonical form of an equivalence class with respect to quasi-homogeneous zero order substitutions into which any first order quasi-homogeneous polynomial with weight (1, 2) should be divided according to the chosen structural principles. It makes sense to reduce to GNF only the systems with different canonical forms in their unperturbed part. The constructive method of resonance equations and sets is used to write the resonance equations with the perturbations of the acquired system satisfying these equations using a formal, almost identical quasi-homogeneous substitution in the original system. The fulfillment of these conditions ensures the formal equivalence of the systems. In addition, the resonance sets of coefficients are specified, which allow obtaining all possible GNF structures and proving that the original system is reducible to a GNF with any of the specified structures. Some examples of characteristic GNFs are presented, including the GNFs with parameter α implying the appearance of an additional resonance equation and the second nonzero coefficient in the appropriate orders of GNFs.

中文翻译：

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拟齐次多项式（$$ \ alpha x_ {1} ^ {2} $$ + x 2，x 1 x 2）的常微分方程组的广义范式。

摘要

本文继续研究广义范式（GNF）的结构排列。考虑在坐标原点进行实数分析的平面系统。它的不受扰动部分形成一阶拟齐次多项式（\（\ alpha x_ {1} ^ {2} \） + x ₂，x ₁ x ₂指定了共振系数集，从而可以获取所有可能的GNF结构，并证明原始系统可还原为具有任何指定结构的GNF。给出了特征性GNF的一些示例，包括参数α暗示着附加谐振方程的出现的GNF和第二个非零系数（以适当的GNF​​阶数出现）。