On the nonexistence of limit cycles of certain quadratic systems (Q1266292)

For the quadratic system: \[ \dot x= -y+\delta x+\ell x^2+ ny^2,\quad \dot y= x(1+ ax- y),\tag{1} \] it is assumed that \[ a< 0,\quad n>1,\quad n+\ell> 0,\quad na^2+\ell< 0,\quad na^2<(n- 1)(\ell+ n)^2.\tag{2} \] It is conjectured that around the anti-saddle \(S_1(x_1,y_1)\) (\(x_1> 0\), \(y_1<1\)) lying on \(1+ ax- y=0\) there exists no limit cycle for any \(\delta\). In \S 1 of the paper, the above conjecture is proved. For the quadratic system: \[ \dot x= -y+ \ell x^2+ mxy+ ny^2,\quad \dot y= x\quad (1+ ax-y),\tag{3} \] let the focal quantities at \(O(0,0)\) satisfy \(W_1= 0\), \(W_2W_3> 0\). It is conjectured that (3) has no limit cycle around \(O\). In \S 3, this conjecture is proved under the conditions \(n>1\), \(\ell<0\). In \S 2, some known results on the nonexistence of limit cycles of quadratic systems obtained by \textit{H. Giacomini}, \textit{J. Llibre} and \textit{M. Viano} [Nonlinearity 9, No. 2, 501-516 (1996; Zbl 0886.58087)] are reproved by more elementary methods.