Chow, Shiu-Nee; Li, Chengzhi; Wang, Duo A simple proof of the uniqueness of periodic orbits in the 1:3 resonance problem. (English) Zbl 0691.58028 Proc. Am. Math. Soc. 105, No. 4, 1025-1032 (1989). 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 Cited in 3 Documents MSC: 37G99 Local and nonlocal bifurcation theory for dynamical systems 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations Keywords:bifurcation diagram; phase portrait; periodic orbit; Picard-Fuchs equation; limit cycles Citations:Zbl 0446.58010; Zbl 0587.34033 PDFBibTeX XMLCite \textit{S.-N. Chow} et al., Proc. Am. Math. Soc. 105, No. 4, 1025--1032 (1989; Zbl 0691.58028) Full Text: DOI