Hilbert's 16th problem for quadratic vector fields
The second part of Hilbert's 16th problem is to determine the number and relative position of the limit cycles of a polynomial vector field in the plane. This problem remains open even for the case of a vector field whose components are quadratic polynomials. Perhaps the most important and deepest result in the direction of Hilbert's problem is the recent solution of the Dulac problem, solved independently by J. Ecalle and Yu. Il'yashenko. In particular, this work has resulted in the following theorem: A polynomial vector field in the plane has at most a finite number of limit cycles. For a history and a proof of this theorem, see [\textit{Yu. Il'yashenko}, Finiteness theorems for limit cycles, Transl. of Mathematical Monographs, Vol. 94, Am. Math. Soc. (1991; Zbl 0743.34036)]. In view of the solution of Dulac's problem, the unresolved portion of Hilbert's problem is usually stated as follows: For each fixed integer \(N\geq 2\), determine the maximum number of limit cycles for a polynomial vector field of degree \(N\). However, as an intermediate result, it also remains to show that, for each fixed degree, there exists a uniform bound for the number of limit cycles. This is the problem addressed in the paper under review for the case of quadratic vector fields. The main idea underlying the paper is that the existence of a uniform bound on the number of limit cycles can be proved from local finiteness results. In particular, it is sufficient to determine the ``cyclicity'' of certain graphics, that is, certain invariant sets formed by rest points and connecting orbits. A familiar example of the cyclicity method is provided by Bautin's theorem. It addresses the case, where the graphic is a single point, namely, a weak focus or a center of a quadratic system. To state this result, let \(X\) denote a quadratic system with a weak focus or a center. The cyclicity of this rest point in the space of quadratic vector fields is the maximum number of limit cycles that can appear near it after an arbitrary quadratic perturbation of \(X\). Bautin's theorem asserts that the cyclicity in this case is at most three. A theorem of the second author [Dynamical systems, Proc. Symp. Valparaiso/Chile 1986, Lect. Notes Math. 1331, 161-168 (1988; Zbl 0676.58046)] states that if, in the class of polynomial vector fields of a fixed degree, each ``graphic'' has finite cyclicity relative to the class, then there is a uniform bound for the number of limit cycles. In the paper under review, the authors apply this result to the class of quadratic systems. In fact, they present a list of 121 graphics that occur in quadratic systems and they prove that if each of the graphics on this list has finite cyclicity, then there exists a uniform bound for the number of limit cycles over the class of quadratic planar vector fields. Their result defines a clear program which leads to a solution of the existential part of Hilbert's problem. The examination of the cyclicity of graphics leads to many interesting and difficult unsolved problems that have interest beyong their connection with Hilbert's problem. The authors of this paper have made important contributions to research in this direction. Their paper should be read by all students of bifurcation theory and especially those interested in the resolution of Hilbert's problem.
number and relative position of the limit cycles
scientific article; zbMATH DE number 599886