Abstract
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 center problem for planar polynomial vectorfields.
Highlights
In this paper we deal with the center problem for the trigonometric Abel equation dρ/dθ = a1(θ)ρ2 + a2(θ)ρ3, where a1(θ) and a2(θ) are trigonometric polynomials in θ
Blinov in [9] proved the following result which shows that the lowest possible degree such that an Abel equation can have a non-universal center is at least 3
In this paper we study the centers of equation (1.1) when a1(θ) and a2(θ) are trigonometric polynomials of degree 3, i.e., a1(θ) = b00 + b10 cos θ + b01 sin θ + b20 cos(2θ) + b02 sin(2θ) +b30 cos(3θ) + b03 sin(3θ), a2(θ) = c00 + c10 cos θ + c01 sin θ + c20 cos(2θ) + c02 sin(2θ) +c30 cos(3θ) + c03 sin(3θ), (2.1)
Summary
In [15] there is another example of a center of an Abel equation which is not universal and where a1(θ) and a2(θ) are trigonometric polynomials of degree 3 and 6 respectively. The following open problem can be established: Open problem: To determine the lowest degree of the trigonometric polynomials a1(θ) and a2(θ) such that the Abel equation (1.1) has a center which is not universal. Blinov in [9] proved the following result which shows that the lowest possible degree such that an Abel equation can have a non-universal center is at least 3.
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