A generalized normal form and formal equivalence of two-dimensional systems with quadratic zero approximation: IV

We continue the study of invertible formal transformations of two-dimensional autonomous systems of differential equations with zero approximation represented by homogeneous polynomials of degree 2 and with perturbations in the form of power series without terms of order < 3. In the regular case, we consider systems that have the canonical form (αx ₁ ² ? sgnα x ₂ ² , x ₁ x ₂) with α ≠ 0 as the zero approximation. For such systems, we obtain resonance equations in closed form and use them to prove the theorem on the formal equivalence of systems and establish a generalized normal form to which any original system can be reduced by an invertible change of variables.