Classification of Quasilinear Systems of Ordinary Differential Equations and Its Application to Normalization of Systems in the Bogdanov–Takens Critical Case

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