The quadratic system having a parabola as its integral curve has at most one limit cycle (Q1320584)

The authors prove in this paper the following Theorem. When a quadratic system has a parabola as its integral curve, then its limit cycle, if exists, must be unique. The normal forms used by them are those of S. P. Chen: (1) \(dx/dt= (xy+ \mu)\), \(dy/dt= (xy+\mu)(x+\ell)+(2y\mp 1)(y-{1\over 2} x^ 2-\ell x-m)\), \(\mu>0\). These forms are transformed into the Liénard forms: (2) \(\dot x= e^ y\pm ({1\over 2} x^ 3+ \ell x^ 2+ mx+ \mu)/x^ 3\), \(\dot y=- (x\pm 2\mu)/x^ 4\) by the transformations \(u= y-{1\over 2} x^ 2-\ell x-m\), \(v=x\) and \(x_ 1= v\), \(y_ 1=\ell n(\pm\mu/v^ 2)\), \(\pm v^ 3 t=\tau\) (\(x_ 1\), \(y_ 1\), \(\tau\) are replaced by \(x\), \(y\), \(t\) in (2) now). Finally, the well-known theorem of Z. F. Zhang is used to prove the uniqueness of limit cycles for systems (2).