Existence of limit cycles for real quadratic differential systems with an invariant cubic (Q868747)

A quadratic system is a system of ordinary differential equations of the form \[ \dot x = P(x,y),\quad \dot y = Q(x,y) (QS), \] where \(P\) and \(Q\) are relatively prime polynomials for which \(\text{max}\{\deg P, \deg Q \} = 2\). An invariant cubic is an algebraic curve \(f(x,y) = 0\), where \(f\) is a cubic polynomial, which is composed of orbits in the phase portrait of system (QS). Any quadratic system that has an irreducible cubic invariant curve can, by an affine change of coordinates and a time rescaling, be placed in one of twelve canonical forms, for each of which the form of \(f\) is known. In this paper the authors show that in exactly two of these classes of canonical forms is a limit cycle in the phase portrait possible, and give an example of a canonical system with a limit cycle in each of these two cases. (It is impossible that the cycle in question actually lie in the cubic curve. The authors give a new and the simplest known proof of this well-known fact.)