Relationships between limit cycles and algebraic invariant curves for quadratic systems (Q852689)

The paper deals with algebraic limit cycles in quadratic systems. As it is well known, seven types of algebraic limit cycles have been found: one of degree 2, four of degree 4, one of degree 5 and one of degree 6. An algebraic limit cycle \(\gamma\) of a planar system \(\dot{x}=p(x,y)\), \(\dot{y}=q(x,y)\) is contained in the zero set of certain invariant algebraic curve \(\varphi(x,y)=0\), i.e., satisfying \(p\partial\varphi/\partial x+ q\partial\varphi/\partial y -k\varphi=0\), \(k\in {\mathbb R}[x,y]\) being the cofactor. Here, the authors prove five theorems on the relationships between \(\gamma\), \(\varphi\) and \(k\) in the case when \(k=0\) is an isocline, or contains a singular point of the system, as well as when \(\varphi=0\) intersects the line at infinity at exactly one point. The results obtained suggest that the following conjecture is relevant: if a quadratic system has an algebraic limit cycle, then it is the unique limit cycle of the system.