Geodesic
Cyclicity of elementary polycycles with fixed number of singular points in generic $k$-parameter families
Algebra i analiz, Tome 22 (2010) no. 4, pp. 57-75.

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An estimate is found for the number of limit cycles arising from polycycles in generic finite-parameter families of differential equations on the two-sphere. It is proved that if the polycycles have a fixed number of singular points and all the singular points are elementary, then an estimate of cyclicity holds true, which is polynomial in the number of parameters of the family.
Keywords: number of limit cycles, polycycle, Hilbert's sixteenth problem, Hilbert–Arnol'd problem. 
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P. I. Kaleda; I. V. Shchurov. Cyclicity of elementary polycycles with fixed number of singular points in generic $k$-parameter families. Algebra i analiz, Tome 22 (2010) no. 4, pp. 57-75. http://geodesic.mathdoc.fr/item/AA_2010_22_4_a2/

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