A trichotomy for generic sectional-hyperbolic chain-recurrent classes

Abstract

The notion of sectional-hyperbolicity is a weakened form of hyperbolicity introduced for vector fields in order to understand the dynamical behavior of certain higher-dimensional systems such as the multidimensional Lorenz attractor. In this paper we address the questions proposed in [\emph{Math. Z.}, \textbf{298} (2021), 469-488] and we provide a partial answer by proving that a $C^1$-generic non-trivial sectional-hyperbolic chain-recurrent class, not necessarily Lyapunov stable, satisfies a trichotomy: it is either a homoclinic loop, a union of saddle connections between singularities, or it is robustly a homoclinic class.