The algebraic limit cycles of planar cubic systems (Q2227025)

Algebraic limit cycles of differential systems of the form \[\dot{x}=x+P_3(x,y),\quad \dot{y}=y+Q_3(x,y), \] where \(P_3(x,y)\) and \(Q_3(x,y)\) are homogeneous cubic polynomials, are studied. Note that the results of Theorem 1 were obtained much earlier in the monograph [\textit{V. N. Gorbuzov} and \textit{A. A. Samodurov}, The Darboux equation and its analogies. Textbook. (Uravnenie Darbu i ego analogi. Uchebnoe posobie) (Russian). Grodno: Grodnenskij Gosudarstvennyj Universitet (1985; Zbl 0607.34001)]. The Darboux equations of a more general form are also considered there. The rest of the results of the article can also be obtained on the basis of this monograph.