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A simple proof of the uniqueness of periodic orbits in the 1:3 resonance problem. (English) Zbl 0691.58028

E. I. Khorozov [Tr. Semin. Im. I. G. Petrovskogo 5, 163-192 (1979; Zbl 0446.58010)] considered the versal deformation of a planar vector field which is invariant under a rotation through an angle \(2\pi\) /3. In his proof of the uniqueness of limit cycles in certain regions, some results from algebraic geometry were applied. In this paper, the authors present a more elementary proof for the uniqueness by using the Picard Fuchs equation and a technique given by J. Carr, S. Chow, H. Hale [J. Differ. Equations 59, 413-436 (1985; Zbl 0587.34033)].
Reviewer: K.Chang

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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