Limit cycles and bifurcation curves for the quadratic differential system \((III)_{m=0}\) having three anti-saddles. I (Q1917810)

The paper is concerned with the quadratic system (1) \(dx/dt = - y + \delta x + \ell x^2 + ny^2\), \(dy/dt = x(1 + ax - y)\), in which \(- 1 < \ell < 0\), \(n + \ell - 1 > 0\). By further assuming that \(na^2 + \ell < 0\), this will ensure that the four critical points of (1) form the vertices of a concave quadrilateral with \((0, {1 \over n})\) being a saddle, and the origin and the two critical points \(S_1 (x_1, y_1)\), \(S_2 (x_2, y_2)\), \((x_2 < 0 < x_1)\), on the line \(y = 1 + ax\) all being anti-saddles. The detailed analysis done in the paper allows the author to draw in the \((a, \delta)\) parameter plane the global bifurcation of trajectories around the origin.