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Monodromic unbounded polycycles. (English) Zbl 0959.34023

R. Conti [Nonlinear oscillations, Proc. 11th Int. Conf., Budapest 1987, 36-43 (1987; Zbl 0627.34032)] has posed the following problem. Let \({\mathcal V}\) be a polynomial vector field in \(\mathbb{R}^2\) of degree \(n\) and \(S\) a center of the flow generated by \({\mathcal V}\). Suppose \(N_S\), the maximal region of closed orbits surrounding \(S\), is unbounded and denote by \(\partial N_S\) its boundary. He conjectured that the \(\partial N_S\) cannot have more than \(n-1\) connected components. In another paper [Arch. Math. Brno, 26, No. 2/3, 93-100 (1990; Zbl 0732.34027)], Conti showed, with examples, that the number of connected components of \(\partial N_S\) is at least \(n-1\), for any \(n\), while it cannot exceed \(n+1\). In this paper, the author proves that the number of there components cannot be more than \(n\).

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
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