Stability conditions for reversible and partially integrable systems

Abstract

The purpose of this paper is to provide results like Peixoto’s Theorem inside a class $\mathfrak{X}$ of 3-dimensional reversible systems. We deal with reversible systems associated to an involution ${\phi }_1$ such that $\mathrm{Dim}\mathrm{Fix}({\phi }_1)$ is equal to one. We further assume the existence of a first integral for any $X\in \mathfrak{X}$ that leaves invariant any sphere centered at the origin. The main result establishes generic necessary conditions for the structural stability in $\mathfrak{X}$. Stability conditions on generic one-parameter families of reversible vector fields on the two-dimensional sphere are also stated.