A counterexample to the composition condition conjecture for polynomial Abel differential equations
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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 pointed out that all centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.
Is part ofErgodic Theory and Dynamical Systems, 2019, vol. 39, núm. 12, p. 3347-3352
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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 ...
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 ...