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The finiteness problem for limit cycles of polynomial vector fields on the plane, germs of saddle resonant vector fields and non-Hausdorff Riemann surfaces. (English) Zbl 0588.34024

Topology, general and algebraic topology, and applications, Proc. int. Conf., Leningrad 1982, Lect. Notes Math. 1060, 290-305 (1984).
[For the entire collection see Zbl 0527.00015.]
The main valid result of Dulac’s paper ”On limit cycles” (Dulac’s theorem) gives the asymptotic expansion up to arbitrary degree of x for monodromy transformation of the separatrix polygon of the analytic vector field in \(R^ 2\). Dulac reduces from this result the finiteness theorem for limit cycles of the polynomial vector field in \(R^ 2\). The mistake in this reduction is mentioned. The modern proof of Dulac’s theorem for smooth vector fields is outlined. This proof makes use of orbital smooth classification for elementary singular points (i.e. the singular points of vector fields in \(R^ 2\) with at least one nonzero eigenvalue of linearisation in this point). This classification is the complete list (obtained by parts by M. Hukuhara, K. T. Chen, F. Takens, A. D. Brjuno, R. I. Bogdanov and others). One simplest finiteness theorem is proved for polynomial vector fields with components from Q(x,y) and nondegenerate singular points. The orbital analytic classification of germs of saddle resonant vector fields (of the form \(px\partial /\partial x+qy\partial /\partial y+...)\) is given; \((x,y)\in C^ 2\). This classification has functional moduli. It was obtained by the author together with S. M. Voronin and P. M. Elisarov, and independently by J. Martinet and J. P. Ramis.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Citations:

Zbl 0527.00015