Alien limit cycles near a Hamiltonian 2-saddle cycle

Abstract

It is known that perturbations from a Hamiltonian 2-saddle cycle Gamma can produce limit cycles that are not covered by the Abelian integral, even when it is generic. These limit cycles are called alien limit cycles. This phenomenon cannot appear in the case that Gamma is a periodic orbit, a non-degenerate singularity, or a saddle loop. In this Note, we present a way to study this phenomenon in a particular unfolding of a Hamiltonian 2-saddle cycle, keeping one connection unbroken at the bifurcation.