The problem of center for resonant singular points of polynomial vector fields (Q1361821)

As is well-known, the general integral of the last center case for a real quadratic field (which can generate three limit cycles after small perturbations) can be obtained by applying to the field a complex linear transformation \(x={1\over 2i} (u-v)\), \(y={1\over 2} (u+v)\) and transform it into the field \[ \dot u= u+ A_1u^2+ B_1uv+ C_1v^2,\quad \dot v=-v+ A_2u^2+ B_2uv+ C_2v^2\tag{1} \] with complex coefficients [see \textit{Y. C. Ye}, Theory of limit cycles, Translations of Mathematical Monographs 66. Providence, Rhode Island (1986; Zbl 0588.34022), \S 9). (1) is called by the author ``having \(O(0,0)\) as a \(1:-1\) resonant center with cyclicity 3''. He shows that the quadratic perturbation of the \(p:-q\) \((0<p<q)\) quadratic resonant saddle: \[ \dot u= pu+ 2pqu^2+ 2pv^2+ \varepsilon_1 u+\varepsilon_2u^2+ \varepsilon_3uv,\;\dot v= -qv+ pquv- \varepsilon_4u^2- \varepsilon_5 v^2 \] (\(\varepsilon_i>0\), small) can have at least four small amplitude limit cycles. Aside from this, he also studies the node case \(1\leq p<-q\) and the saddle node case \(p=1\), \(q=0\) for the general polynomial system: \[ \dot x= px+\text{h.o.t.},\quad \dot y=-qy+\text{h.o.t.}. \]