Classification of Quasilinear Systems of Ordinary DifferentialEquations and Its Application to Normalization of Systems in the Bogdanov–Takens CriticalCase

We consider a two-dimensional autonomous system with a first-order weight $$(1,2) $$ quasihomogeneous polynomial in the unperturbed part. For the unperturbed part, we provide a classification according to which the set of such polynomials is constructively divided into eight equivalence classes with respect to zero-order quasihomogeneous changes of variables. In each class, we single out a representative, referred to as a canonical form. All structures of generalized normal forms are obtained for the remaining unstudied system with one of the canonical forms in the unperturbed part. The system with unperturbed part $$ (x_2,ax_1x_2+bx_1^3)$$ is normalized by the method of resonance equations and sets, which dramatically improves the results already obtained in the research of one of the critical cases of the Bogdanov–Takens classification.