Dulac-Cherkas functions for generalized Liénard systems

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 Liénard systems of the type $ \frac{dx}{dt} = y, \quad \frac{dy}{dt} = \sum_{j=0}^l h_j(x) y^j$ with $l \ge 1$. In case $1 \le l \le 3$, linear differential equations play a key role in this process, for $ l \ge 4$, we have to solve a system of linear differential and algebraic equations, where the number of equations is larger than the number of unknowns. Finally, we show that Dulac-Cherkas functions can be used to construct generalized Liénard systems with any $l$ possessing limit cycles.