Yakovenko, S. A geometric proof of the Bautin theorem. (English) Zbl 0828.34026 Ilyashenko, Yu. (ed.) et al., Concerning the Hilbert 16th problem. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 165(23), 203-219 (1995). A new proof of the Bautin theorem on bifurcations of limit cycles from an elliptic singular point is given. Making use of the hidden \(Z_3\)- symmetry of an auxiliary Hamiltonian system the author avoids lengthy computations exploited in the original proof and arrives to the following reformulation of Bautin’s result: Theorem. Cyclicity, within the family of quadratic vector fields, of a polycycle consisting of just one elliptic point, equals 3.For the entire collection see [Zbl 0819.00005]. Reviewer: Yu.V.Rogovchenko (Firenze) Cited in 29 Documents MSC: 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations Keywords:cyclicity; Bautin theorem; bifurcations of limit cycles; elliptic singular point; quadratic vector fields; polycycle PDFBibTeX XMLCite \textit{S. Yakovenko}, Transl., Ser. 2, Am. Math. Soc. 165, 203--219 (1995; Zbl 0828.34026)