On the number of critical periods for planar polynomial systems of arbitrary degree
A first approach to the celebrate Hilbert's 16th problem is the construction of examples having as many limit cycles as possible, in other words to find lower bounds for the so-called Hilbert's numbers. If one only considers limit cycles surrounding one critical point, then the best lower bound for the number of limit cycles is \(O(n^2)\), while the best results concerning any configuration of limit cycles provide \(O( n^2 \log n)\) lower bounds. It is worth to notice the parallelism between Hilbert's 16th problem and the problem of the number of critical period of a planar vector field of degree \(n\). In this paper the authors construct a class of planar systems of arbitrary degree \(n\) having a reversible center at the origin and such that the number of critical periods on its period annulus grows quadratically with \(n\). Previous results on this subject generated systems having linear growth.
scientific article; zbMATH DE number 5757644