Local dynamics of some algebraic maps. (English) Zbl 0946.34026
H. Zoladek [J. Differ. Equations 137, No. 1, 94-118 (1997; Zbl 0885.34034)] stated a nontrivial algebraic center-focus problem for the following implicitly defined map \(z\to w\) with
\[
w+ z+ \sum^n_{i+ j= 2} a_{ij} z^i w^i= 0.\tag{1}
\]
The equation has an analytic solution of the form \(w=\widetilde f(z)= -z+\cdots\). In this paper, the authors study the map \(\widetilde f\) where \(\widetilde f\) is defined through the cubic case of equation (1) with \(n= 3\). They use algebraic tools, based on the Gröbner bases theory, and Lyapunov function methods to examine the center-focus problem and the problem of estimating the number of limit cycles near \(z= 0\).
Reviewer: A.-A.Yakubu (Washington/D.C.)
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C08 | Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) |
