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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 1 x 2 +B 1 xy+Cy 2 ) x +(-2y+Dx 2 +A 2 xy+B 2 y 2 ) y ·

There are twenty cases of center. Their necessity is proved by 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”.


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