Limit cycles and bifurcation curves for the quadratic differential system (III)\(_{m=0}\) having three anti-saddles (II) (Q1364963)

In the author's paper \((*)\) [Chin. Ann. Math., Ser. B 17, No. 2, 167-174 (1996; Zbl 0855.34035)], he studied the bifurcation problem of the quadratic system \[ (\text{III})_{m= 0}: \dot x= -y+\delta x+\ell x^2+ ny^2,\quad \dot y= x(1+ ax-y) \] around the focus \(0(0,0)\) under the conditions \(m= 0\), \(-1<\ell<0\), \(b= -1\), \(n+\ell- 1>0\), \(a\leq 0\), and stated the bifurcation diagram in the \((a,\delta)\) plane. As a continuation of the paper \((*)\), this paper studies the limit cycle bifurcation around the focus \(S_1(x_0, y_0)\) lying on \(y= 1+ax\) as \(\delta\) varies, where \(x_0>0\), \(y_0>0\). A conjecture on the nonexistence of limit cycles around \(S_1\) and another one on the nonexistence of limit cycles around both \(0\) and \(S_1\) are given, together with some numerical examples. The author claims that the results are very interesting but they are not so satisfactory as in the paper \((*)\).