Abstract
In this work we consider the polynomial differential system $\dot x = -y + x y^{n-1}$, $\dot y =x + a y x^{n-1}$, where $a \in \mathbb{R}$ and $n \ge 2$ with $n \in \mathbb{N}$. This system is a certain generalization of the classical Liénard system. We study the center problem and consequently the analytic integrability problem for such family around the origin for any value of $n$.
Highlights
Two of the main problems in the qualitative theory of differential systems are the center/focus problem and the integrability problem that are equivalent for systems with a linear part of center type
For other singularities the existence of analytic invariant curves is connected with the analytic integrability and the existence of a explicit first integral, see for instance [1] and references therein
Despite the intense activity on the well-known center/focus problem, there are very few satisfactory results on characterizing whether a given finite singular point is a center or a weak focus for any polynomial system in function of its degree, see for instance [4,5,6, 26]. This is mainly due to the fact that most of the results on the center/focus problem has been done considering particular differential systems because the computations of the focal values are very involved needing a computer algebra assistance, in the majority of the cases
Summary
Two of the main problems in the qualitative theory of differential systems are the center/focus problem and the integrability problem that are equivalent for systems with a linear part of center type. This is mainly due to the fact that most of the results on the center/focus problem has been done considering particular differential systems because the computations of the focal values (see below for a definition) are very involved needing a computer algebra assistance, in the majority of the cases. In the last decades several generalizations of the Liénard equations have been proposed, see for instance [2, 5, 7, 16,17,18, 23, 24, 27] where the authors studied the center problem and the number of limit cycles that bifurcate from the singular point at the origin.
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