Yu. S. Ilyashenko, A. Panov, “Some upper estimates of the number of limit cycles of planar vector fields with applications to Liénard equations”, Mosc. Math. J., 2001, Volume 1, Number 4,Pages <nobr>583

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Some upper estimates of the number of limit cycles of planar vector fields with applications to Liénard equations

Yu. S. Ilyashenko^abcd, A. Panov^ab

^a M. V. Lomonosov Moscow State University
^b Independent University of Moscow
^c Steklov Mathematical Institute, Russian Academy of Sciences
^d Cornell University

Abstract: We estimate the number of limit cycles of planar vector fields through the size of the domain of the Poincaré map, the increment of this map, and the width of the complex domain to which the Poincaré map may be analytically extended. The estimate is based on the relationship between the growth and zeros of holomorphic functions [IYa], [I]. This estimate is then applied to getting the upper bound of the number of limit cycles of the Liénard equation $\dot x=y-F(x)$, $\dot y=-x$ through the (odd) power of the monic polynomial $F$ and magnitudes of its coefficients.

Key words and phrases: Limit cycles, Poincarй map, Liйnard equation.

MSC: 34Cxx, 34Mxx

Received: October 30, 2001; in revised form December 19, 2001

Language: English

DOI: 10.17323/1609-4514-2001-1-4-583-599