Generalized Normal Forms of the Systems of Ordinary Differential Equations with a Quasi-Homogeneous Polynomial ($$\alpha x_{1}^{2}$$ + x 2, x 1 x 2) in the Unperturbed Part

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.