A trichotomy for generic sectional-hyperbolic chain-recurrent classes

TL;DR

This paper establishes a trichotomy for generic sectional-hyperbolic chain-recurrent classes, classifying their structure into three distinct types, which advances understanding of complex dynamical systems like the Lorenz attractor.

Contribution

It proves that generic non-trivial sectional-hyperbolic chain-recurrent classes satisfy a trichotomy, clarifying their possible configurations in higher-dimensional systems.

Findings

Classified chain-recurrent classes into three types: homoclinic loops, saddle connection unions, or robust homoclinic classes.

Extended understanding of the structure of sectional-hyperbolic systems.

Provides partial answers to previously posed questions in the field.

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.