Limit Cycles for Multidimensional Vector Fields. The Elliptic Case

We consider polynomial vector fields of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\dot x = 2y + zR(x,y), \dot y = 3x^2 - 3 + z{\text{S}}(x{\text{,}}y),{\text{ }}\dot z = A(x,y)z,{\text{ }}z \in \mathbb{R}^v $$ \end{document} and their polynomial perturbations of degree ≤n. 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 y2 − x3 + 3x = h, leads to a certain 2- dimensional integral (which generalizes the elliptic integrals). We show that this integral has a representation R1(h)I1 + ⋅⋅⋅ + Re(h)Ie, where Rj(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.