Abstract
We consider the Abel Equationρ′ =p(θ)ρ 2 +q(θ)ρ 3 (*) withp (θ),q(θ) polynomials in sin θ, cos θ. The center problem for this equation (which is closely related to the classical center problem for polynomial vector fields on the plane) is to find conditions onp andq under which all the solutions р(θ) of this equation are periodic, i.e. р(0)=р(2π) for all initial values р(0). We consider the equation (*) as an equation on the complex plane\(\frac{{dy}}{{dz}} = p(z)y^2 + q(z)y^3 \) (**) withp, q_— Laurent polynomials. Then the center condition is that its solutiony(z) is a univalued function along the circle |z|=1. We study the behavior of the equation (**) under mappings of the complex plane onto Riimann Surfaces. This approach relates the center problem to the algebra of rational functions under composition and to the geometry of rational curves We obtain the sufficient conditions for the center in the form ∫|z|=1 P i Q j dP=0 withP=∫p,Q=∫q.
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Submitted by J. P. Françoise
The research was supported by the Israel Science Foundation (Grant No. 117/99-1) and by the Minerva Foundation.
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Blinov, M., Yomdin, Y. Center and composition conditions for Abel differential equation, and rational curves. Qual. Th. Dyn. Syst. 2, 111–127 (2001). https://doi.org/10.1007/BF02969385
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DOI: https://doi.org/10.1007/BF02969385