Universal centers in the cubic trigonometric Abel equation
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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.
Is part ofElectronic Journal of Qualitative Theory of Differential Equations, 2014, núm. 1, p. 1-7
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Giné, Jaume; Grau Montaña, Maite; Santallusia Esvert, Xavier (Shanghai Normal University & Wilmington Scientific Publisher, 2013)In this paper we deal with the center problem for the trigonometricAbel equation dρ/dρ =a1(θ)ρ^2+a2(θ)ρ^3; where a1(θ) and a2(θ)are trigonometric polynomials in θ. This problem is closely connectedwith the classical Poincar´e ...
Giné, Jaume; Grau Montaña, Maite; Santallusia Esvert, Xavier (Elsevier, 2015)The classical Poincaré center-focus problem for planar polynomial systems of ordinary differential equations can be transformed, in certain particular cases, to the center problem of a trigonometric Abel differential ...
A counterexample to the composition condition conjecture for polynomial Abel differential equations Giné, Jaume; Grau Montaña, Maite; Santallusia Esvert, Xavier (Cambridge University Press, 2019)Polynomial Abel differential equations are considered a model problem for the classical Poincaré center–focus problem for planar polynomial systems of ordinary differential equations. In the last few decades, several works ...