On the construction of a class of Dulac-Cherkas functions for generalized Li\'enard systems

Leonid Cherkas , Alexander Grin , Klaus Schneider

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 8, 1-20

MSC 2000

34C07 Theory of limit cycles of polynomial and analytic vector fields

34C05 Location of integral curves, singular points, limit cycles

Abstract
Dulac-Cherkas functions can be used to derive an upper bound for the number of limit cycles of planar autonomous differential systems including criteria for the non-existence of limit cycles, at the same time they provide information about their stability and hyperbolicity. In this paper, we present a method to construct a special class of Dulac-Cherkas functions for generalized Li\'enard systems of the type $ \frac{dx}{dt} = y, \quad \frac{dy}{dt} = \sum_{j=0}^l h_j(x) y^j$ with $l \ge 1$ by means of linear differential equations. In case $1 \le l \le 3$, the described algorithm works generically. We show that this approach can be applied also to systems with $l \ge 4$. Additionally, we show that Dulac-Cherkas functions can be used to construct generalized Li\'enard systems with any $l$ possessing limit cycles.