Quadratic systems possessing a weak focus of order two and a singular point with a zero characteristic root (Q921219)

The following quadratic system possessing a weak focus of order two and a singular point with a zero characteristic root is considered: \[ (1)\quad \frac{dx}{dt}=-y+\ell x^ 2+mxy+y^ 2,\quad \frac{dy}{dt}=x(1+ax- y),\quad a>0, \] where \(m=\frac{a(z\ell -1)}{\ell +1}\), \(\ell \neq -2,- 1,0,1/2\). Theorem. System (1) (A) has no limit cycle when \(\ell \in (- \infty,+\infty)\setminus (-2,-1)\); (B) has at most one limit cycle when \(\ell \in (-2,-1)\), and the limit cycle must surround the weak focus of order two if it exists; (C) has exactly one limit cycle when \(0<\ell +2\ll 1\). This work is different from that of \textit{W. A. Coppel} [Bull. Aust. Math. Soc., 38, No.1, 1-10 (1988; Zbl 0634.34013)], where a quadratic system possessing a singular point with two zero characteristic roots is considered.