A counterexample to a multidimensional version of the weakened Hilbert's 16-th problem

Abstract

In the weakened 16th Hilbert's Problem one asks for a bound of the number of limit cycles which appear after a polynomial perturbation of a planar polynomial Hamiltonian vector field. It is known that this number is finite for an individual vector field. In the multidimensional generalization of this problem one considers polynomial perturbation of a polynomial vector field with invariant plane supporting a Hamiltonian dynamics. We present an explicit example of such perturbation with infinite number of limit cycles which accumulate at some separatrix loop.