Centers: their integrability and relations with the divergence

Abstract

Abstract. This is a brief survey on the centers of the analytic differential systems in R 2 . First we consider the kind of integrability of the different types of centers, and after we analyze the focus-center problem, i.e. how to distinguish a center from a focus. This is a difficult problem which is not completely solved. We shall present some recent results using the divergence of the differential system.

Global Analysis of a Liénard System with Quadratic Damping

“…All kinds of problems about Liénard equations are always the focus of the theory of differential equations. In 2016, Llibre [10] studied the centers of the analytic differential systems and analyzed the focus-center problem. H. Chen and X. Chen [11][12][13] investigated the dynamical behaviour of a cubic Liénard system with global parameters, analyzing the qualitative properties of all the equilibria and judging the existence of limit cycles and homoclinic loops for the whole parameter plane.…”

Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems

“…Moreover, in a neighborhood of a linear center of an analytic system always exists a local analytic first integral, as it was proven by Poincaré [29] and Lyapunov [22]. In general there do not exist local analytic first integrals in the neighborhoods of the nilpotent or degenerate centers, see for instance [21]. But any center always has a local C ∞ first integral, see [24].…”

“…Finally, the degenerate centers are the centers whose linear part is identically zero. For more details on these three kinds of centers see for instance [21] and the references quoted there.…”

“…For analytic and in particular for polynomial differential systems, there are some methods for studying the nilpotent centers, see [12,16,21,25,27]. But the determination of the degenerate centers of the analytical differential systems is more difficult and only very partial results have been obtained.…”

“…Obviously we only need to consider the reduced systems (19)- (21) for the existence of centers at the origin. For system (21) we compute G + 3 (θ) = s 1ẏ cos θ − s 2ẋ sin θ = 5 cos 3 θ − 3 sin 5 θ, according to (8). It follows that G + 3 (θ) has a real root θ ≈ 0.93943 for θ ∈ [0, π].…”