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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].

MSC:

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