The center problem and composition condition for Abel differential equations
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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 equation. Several research papers focused on the study of the center problem for trigonometric Abel differential equations. Polynomial Abel differential equations are also considered in the literature as a model problem. In this work we make a survey of the most important results in this context and we provide the state of the art of several related conjectures. We give two new results on these conjectures.
Is part ofExpositiones Mathematicae, 2016, vol. 34, núm. 2, p. 210-222
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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 ...
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; Valls, Claudia (Elsevier, 2017)In this paper we study the center problem for Abel polynomial differential equations of second kind. Computing the focal values and using modular arithmetics and Gröbner bases we find the center conditions for such systems ...