On a method of lyapunov quantities computation for some Lienard systems

Background. The problem of finding the maximum number of limit cycles arising in the differential equation of the first order is the second part of the 16th Hilbert problem. It has been of constant interest to mathematicians for more than 100 years. And although some particular results of solving this problem are known, it has not yet been fully resolved. The aim of this paper is the practical implementation of one of the methods for calculating Lyapunov quantities, which was described in general terms in the papers of Lloyd and Lynch ([5, 6]). The method is used to estimate the maximum number of small-amplitude limit cycles in some Lienard systems (equations). Materials ans methods. Lloyd and Lynch proved that when the right-hand sides of the Lienard system are expanded in Taylor series, some relation depends on the parameter k. This parameter is directly related to the possible number of small-amplitude limit cycles arising in the system. We propose a procedure for the exact determination of the function $F^*(u)$ (the right-hand side of the equation) in the form of a series whose terms are determined using the representation in the form of Bell polynomials, according to the formula of Faa di Bruno. Results. A formula is obtained which makes it possible to find Lyapunov quantities of arbitrary order for certain Lienard systems up to a negative factor. The calculations are compared with known formulas and the applicability of the proposed method for estimating the number of small-amplitude limit cycles in the Lienard system is shown. Conclusions. The technical realization of the method described in [6] is performed, which makes it possible to easily find Lyapunov quantities, which makes it possible to estimate the maximum number of small-amplitude limit cycles arising from the fixed point of the Lienard system.