Limit cycles for multidimensional vector fields. The elliptic case

We consider polynomial vector fields of the form (x) over dot = 2y + zR( x, y), (y) over dot = 3x(2) - 3 + zS(x, y), (z) over dot = A(x, y)z, z is an element of R-nu, and their polynomial perturbations of degree less than or equal ton. We present a sufficient condition that the perturbed system has an invariant surface close to the plane z = 0. We study limit cycles which appear on this surface. The linearized condition for limit cycles, bifurcating from the curves y(2) - x(3) + 3x = h, leads to a certain 2-dimensional integral (which generalizes the elliptic integrals). We show that this integral has a rep resentation R-1(h)I-1 + . . . + R-e(h)I-e, where R-j(h) are rational functions with degrees of numerators and denominators bounded by O(n). In the case of constant and one-dimensional matrix A(x, y) we estimate the number of zeros of the integral by const .n.