Alien limit cycles near a Hamiltonian 2-saddle cycle
The authors consider smooth families of planar vector fields perturbing the corresponding Hamiltonian system whose phase portrait contains a 2-saddle cycle \(\Gamma\) in the boundary of a continuum of periodic orbits. They claim that perturbations from a Hamiltonian 2-saddle cycle \(\Gamma\) can produce limit cycles which are not covered by the Abelian integrals, even when it is generic, and call these limit cycles as alien limit cycles. The authors assume that such phenomenon cannot appear in the case when \(\Gamma\) is a periodic orbit, a nondegenerate singularity, or a saddle loop and present in their small note a way to study this phenomenon in a particular unfolding of a Hamiltonian 2-saddle cycle keeping one connection unbroken at the bifurcation.
scientific article; zbMATH DE number 2175624