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Quadratic vector fields in the plane have a finite number of limit cycles. (English) Zbl 0625.58028

This paper is an important contribution to Hilbert’s 16-th problem. Hilbert conjectured that any polynomial vector field in the plane has a finite number of limit cycles (isolated periodic orbits). Despite the pioneering work of Dulac (1923), which unfortunately contained an error found in 1985 by Il’yashenko, the problem had remained unsolved for almost a century (until 1987, see below).
Even though Dulac’s proof of the fact that all graphs (closed curves composed of finite number of properly oriented separatrices and singular points, also at infinity) are finite, i.e. do not accumulate limit cycles, failed, his main ideas have persisted and formed the basis for Il’yashenko’s proof of the finiteness property of hyperbolic graphs (containing only hyperbolic points).
The present work completely solves Hilbert’s 16-th problem for quadratic vector fields. They are shown to have always a finite number of limit cycles. Only non-hyperbolic graphs are considered, since for hyperbolic ones Il’yashenko’s theorem is used. All the other cases are dealed with in a completely straightforward way, which represents the big value of this paper. This is also why its value has not been diminished by the recent (1987) proof of the total Hilbert 16-th problem concerning limit cycles. The positive answer to the total Hilbert conjecture was given by J. Ecalle, J. Martinet, R. Moussu and J.-P. Ramis [C. R. Acad. Sci., Paris, Sér. I 304, 375-377; 431-434 (1987; Zbl 0615.58011)], but their proof, despite its beauty, is very difficult to follow, since it uses very advanced and sophisticated techniques.
Reviewer: Z.Denkowska

MSC:

37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems

Citations:

Zbl 0615.58011
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References:

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