The homoclinic structure of generic sectional-hyperbolic classes

Abstract: The notion of sectional-hyperbolicity is a weakened form of hyperbolicity introduced for vector fields in order to understand the dynamic behavior of certain higher-dimensional systems such as the multidimensional Lorenz attractor. In this paper we explore the questions proposed in [\emph{Math. Z.}, \textbf{298} (2021), 469-488] and we provide an affirmative answer by proving that $C^1$-generic non-trivial sectional-hyperbolic chain-recurrent class is robustly a homoclinic class. We then apply our findings to the dynamics of star flows providing a positive answer to Conjecture 1.2 in [\emph{Tran. Amer. Math. Soc.} \textbf{376} (2023), 6845-6871]. We remark that our arguments do not assume Lyapunov stability.