Abstract

In 1998 S. Smale (see [S]) suggested to consider a restriction of the second part of the 16th Hilbert problem to a special class of polynomial vector fields on the plane. This class is called Liénard equations:
x′=y − F (x),
                            (1)
y′= −x.

In 1999 Yu. Ilyashenko and A. Panov (in [IP]) got an explicit upper estimate for the number of limit cycles of Liénard equations through the (odd) power of the monic polynomial F and magnitudes of its coefficients. Their result reclined on the theorem of Ilyashenko and Yakovenko that binds the number of zeros and the growth of a holomorphic function [IYa].

Some years later Yu. Ilyashenko suggested to find an analogue of his joint result with A. Panov for generalized Liénard equations:
x′=yH(x) − xF (x),
                            (2)
y′=−x.

Our main result is an explicit upper bound for a class of generalized Liénard equations, which we called odd. There is a simple pure geometrical condition for an equation to be odd: there would be no limit cycles inside some neighborhood of the infinity (in other words, all limit cycles consists in a neighborhood around the origin). This is direct generalization of the difference between Liénard equations of odd and even degree (see [LMP], [IP] and [CD]).

Recently M. Caubergh and F. Dumortier in [CD] found a linear upper bound for the number of limit cycles in some neighborhood around infinity for even degree Liénard equations. Using it we got an upper estimate for the number of limit cycles of even degree Liénard equations in the focus case.