CLASSIFICATION OF THE QUASILINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND ITS APPLICATION FOR NORMALIZATION OF SYSTEMS IN CRITICAL CASE OF BOGDANOV–TAKENS

Two-dimensional autonomous system with a quasihomogenous polynomial of a first degree and weight (1, 2) in an unperturbed part is considered. The classification of an unperturbed parts is provided. According to it, the set of polynomials is constructively divided into eight equivalence classes with respect to quasihomogenuos zero degree substitutions. In each class the respresentatives, called the canonical forms, are determined. All structures of the generalized normal forms for the so far unstudied system with one of the canonical forms in its unperturbed part are obtained. Normalization in the system with unperturbed part (𝑥2, 𝑎𝑥1𝑥2+𝑏𝑥31) is performed using method of the resonant equations and sets. This significantly improves the already obtained results of the research in one of the critical cases of Bogdanov–Takens classification.