Hamiltonian stability in some open surfaces with simple singularities

Abstract

We complement to some open surfaces with singularities the basic characterization theorem obtained by Jarque--Nitecki [Ergod. Th. \& Dynam Sys. \textbf{20} (2000) 775--799] for Hamiltonian flows in the Euclidean plane which are structurally stable among Hamiltonian flows. We also describe the Hamiltonian stability on $ C=\{x^2+y^2=z^2\}$ (Cone) and $ D=\{xy=0\}$ (Double Crossing) by presenting some characterization theorems for Hamiltonian stability and some natural consequences.