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Solution of the 1: -2 resonant center problem in the quadratic case. (English) Zbl 0943.34018
Authors’ abstract: “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 $$(x+A_1x^2+B_1xy+Cy^2)\partial_x+(-2y+Dx^2+A_2xy+B_2y^2)\partial_y.$$ There are twenty cases of center. Their necessity is proved by {\it A. Fronville} [Algorithmic approach to the center problem for $1:-2$ resonant singular points of polynomial vector fields, Nonlinearity, submitted; not yet available for Zbl ] using factorization of polynomials with integer coefficients modulo prime numbers. Here, it is shown that, in each of the twenty cases, there is an analytic first integral. The authors develop a new method of investigation of analytic properties of polynomial vector fields”.

34C05Location of integral curves, singular points, limit cycles (ODE)
37C20Generic properties, structural stability
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