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Solution of the 1 : −2 resonant center problem in the quadratic case

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Fronville A., Sadovski A. P., Żołądek H.

Fundamenta Mathematicae

1998

tom 157

nr 2-3

191-207

The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field $(x + A_1x^2 + B_1xy + Cy^2) ∂_x+(-2y + Dx^2 + A_2xy + B_2y^2)∂_y$. There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic properties of polynomial vector fields.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Fronville A.

1 Sadovski A. P.

1 Żołądek H.

1 1998

Wyniki wyszukiwania

Solution of the 1 : −2 resonant center problem in the quadratic case