A generalized normal form and formal equivalence of two-dimensional systems with quadratic zero approximation. I (Q1778148)

The paper is a continuation of the paper by \textit{V. V. Basov} [Differ. Equ. 39, 165--181 (2003); translation from Differ. Uravn. 39, No.~2, 145--170 (2003; Zbl 1078.34018)] and studies the problem of formal equivalence of two-dimensional systems with quadratic zero approximation \[ \dot x_i=P_i(x)+X_i(x),\quad i=1,2, \tag{1} \] where \(x=(x_1,x_2)\), \(P_i=a_ix_1^2+2b_ix_1x_2+c_ix_2^2\), \(X_i=\sum_{p=2}^{\infty}X_i^{(p+1)}(x)\) is a perturbation, and \(X_i^{(r)}=\sum_{s=0}^rX_i^{(s,r-s)}x_1^sx_2^{r-s}\). For system (1), resonance equations and a (formal) generalized normal form are defined in a natural way. Along with system (1), the perturbation-free system \[ \dot x_i=P_i(x_1,x_2),\quad i=1,2, \tag{2} \] is considered. It is shown that by a linear nondegenerate change of variables, system (2) can be reduced to one of canonical forms. For the following canonical forms of the zero approximation: 1) \(P_1=P_2=x_1x_2\); 2) \(P_1=P_2=x_1^2\); 3) \(P_1=\alpha x_1^2\), \(P_2=x_1x_2\); 4) \(P_1=x_1^2+\alpha x_2^2\), \(P_2=x_2^2\), generalized normal forms of system (1) are obtained.