Bifurcation and distribution of limit cycles for quadratic differential systems of type II (Q1290954)

The author answers an open problem given by the referee in [Theory of limit cycles. Translations of Mathematical Monographs, Vol. 66, American Mathematical Society (1986; Zbl 0588.34022)] \S 14 about the number of limit cycles around each of the foci \(O(0,0)\) and \(R\left(-{1\over a},y_R\right)\) for the system: \[ \dot x= -y+\delta x+ \ell x^2+ mxy+ ny^2,\quad \dot y= x(1+ ax-y),\quad 0<n<1,\quad a<0, \] when \(\ell= 0\), \(m>-a\), showing that a limit cycle around \(O\) must be unique, but around \(R\) two limit cycles can appear. The author discusses the bifurcation phenomena in detail when \(\ell\neq 0\) (with many phase-portraits in the \((x,y)\)-plane and bifurcation diagrams in the \((\delta,m)\)-plane).