A transfer principle in the real plane from nonsingular algebraic curves to polynomial vector fields
The paper under review is devoted to present two short and elegant proofs of the following Theorem. Let \(f\in\mathbb{R} [X,Y]\) be a polynomial whose gradient \((f_X,f_Y)\) has no zeroes on the zero set \(C\) of \(f\). Then, the planar polynomial vector field \(\dot X=-f_Y -ff_X\); \(\dot Y=-f_X-ff_Y\) has the components of \(C\) as attractors and, in particular, the ovals of \(C\) as the only attracting limit cycles. The first proof uses Duff's rotated vector fields, and the second some ideas coming from Lyapunov's stability theory. This result seems to be the first explicit (and very simple) way to associate to every real algebraic curve of degree \(m\) a polynomial vector field of degree \(2m-1\) having, exactly, the ovals of \(C\) as attracting limit cycles, and it is a nice contribution to the understanding of the interplay between the so-called ``algebraic'' and ``dynamical'' parts of Hilbert's 16th problem. In the article, which contains a short introduction to both parts of the problem, it is also conjectured that the degree \(2m-1\) obtained in the theorem is, in some sense, the smallest one can get for \(4\leq m\).
scientific article; zbMATH DE number 1423185