Uniqueness of limit cycles of quadratic system (III)\(_{m=0}\) (Q1368193)

By using a theorem on the uniqueness of limit cycles due to W. A. Coppel the authors prove the following theorem: Under the conditions: \[ a<0,\;\ell> {1\over 2},\;0<n<1,\;\delta<0 \] the quadratic system \(\text{(III)}_{m=0}\) (Chinese classification) \[ \dot x=-y +\delta x+ \ell x^2+ny^2, \quad \dot y= x(1+ax-y) \] has at most one limit cycle. If it exists, it must be hyperbolic and unstable.