A generalized normal form and formal equivalence of two-dimensional systems with quadratic zero approximation. III. (Q2461877)

The author derives normal forms for the planar systems \[ \begin{aligned} \dot x_1 &=x^2_1+ X_1(x_1,x_2),\;\dot x_2= x^2_1+ x_1 x_2+ X_2(x_1, x_2),\\ \dot x_1 &=-x^2_1- x_1 x_2+ X_1(x_1, x_2),\;\dot x_2= x^2_1+ x_1x_2+ X_2(x_1, x_2),\\ \dot x_1 &=x^2_2+ X_1(x_1,x_2),\;\dot x_2= x^2_1+ X_2(x_1,x_2),\end{aligned} \] where \(X_1\) and \(X_2\) satisfy \(O(|x_1|^3+ |x_2|^3)\) as \(|x_1|+ |x_2|\) tends to zero. For Part I, cf. [\textit{V. V. Basov} and \textit{A. V. Skitovich}, ibid. 39, No. 8, 1067--1081 (2003); translation from Differ. Uravn. 39, No. 8, 1016--1029 (2003; Zbl 1084.34041)]; for Part II [ibid. 41, No. 8, 1061--1074 (2005); translation from Differ. Uravn. 41, No. 8, 1011--1023 (2005; Zbl 1148.34312)].