We summarize known criteria for the non-existence, existence and on the number of limit cycles of autonomous real planar polynomial differential systems, and also provide new results.
We give examples of systems which realize the maximum number of limit cycles provided by each criterion. In particular we consider the class of differential systems of the form ̇x = Pn (x, y) + Pm (x, y), ̇y = Qn (x, y) + Qm (x, y), where n, m are natural numbers with m > n ≥ 1 and (Pi , Qi ) for i = n, m, are quasi-homogeneous vector fields.

(Bolyai Institute. University of Szeged, 2014) Giné, Jaume; Grau Montaña, Maite; Santallusia Esvert, Xavier

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We study the center problem for the trigonometric Abel equation dρ/dθ=a1(θ)ρ2+a2(θ)ρ3,dρ/dθ=a1(θ)ρ2+a2(θ)ρ3, where a1(θ)a1(θ) and a2(θ)a2(θ) are cubic trigonometric polynomials in θθ. This problem is closely connected with the classical Poincaré center problem for planar polynomial vector fields. A particular class of centers, the so-called universal centers or composition centers, is taken into account. An example of non-universal center and a characterization of all the universal centers for such equation are provided.