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The reduced Bautin index of planar vector fields. (English) Zbl 0947.34013
The determination of the number of small-amplitude limit cycles that bifurcate from a weak center of a planar polynomial system is a difficult problem that has only been solved for the case of quadratic systems (by N. N. Bautin). The basic approach to this problem starts with the displacement function $$S$$ defined on a ray emanating from the weak center. If the center is taken to lie at the origin, then $$S$$ is represented by a power series $$\sum_{k=0}^\infty a_k(z)X^k$$ where $$X$$ is the variable along the ray (usually taken to be a coordinate axis) and $$z$$ is the vector variable of coefficients in the polynomial vector field. Because the ring of polynomials is Noetherian, there is a smallest integer $$d$$, called the Bautin index, such that every series coefficient is in the ideal generated by $$\{a_0,a_1,\ldots,a_d\}$$. The maximum number of small-amplitude limit cycles can generally be determined from the value of $$d$$. For example, for the case of quadratic systems, $$d=7$$ and the number of small-amplitude limit cycles is three. In the paper under review, the coefficients of the series expansion for the displacement function are proved to have a special growth property that implies, for each fixed $$z$$, the existence of the positive constants $$\alpha$$ and $$\beta$$ such that the number of limit cycles can be bounded in a disk of the radius $$\alpha (1+|z|)^{-\beta}$$. This bound is given in terms of the reduced Bautin index; that is, the smallest $$\bar d$$ such that every series coefficient is in the integral closure of the ideal generated by $$\{a_0,a_1,\ldots,a_{\bar {d}}\}$$. Although the results in the paper, which are obtained using a sophisticated blend of algebra and analysis, do not provide a bound for the number of small-amplitude limit cycles as a function of the degree of the polynomial vector field, they are a substantial contribution to the resolution of this intriguing problem.

##### 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 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 37C10 Dynamics induced by flows and semiflows 13E05 Commutative Noetherian rings and modules
##### Keywords:
Hilbert’s 16th problem; Bautin index; integral closure
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