On Chaotic Behavior of ASH Attractors

TL;DR

This paper investigates the chaotic properties of ASH attractors, showing they are rescaling-expansive and sensitive to initial conditions, thus contributing to understanding their complex dynamical behavior.

Contribution

It proves that ASH attractors in 3D systems are rescaling-expansive and exhibit sensitivity to initial conditions, extending the understanding of their chaotic nature.

Findings

ASH attractors are rescaling-expansive.

ASH attractors show sensitivity to initial conditions.

The results apply to $C^1$ vector fields on 3D manifolds.

Abstract

The asymptotic sectional hyperbolicity is a weak notion of hyperbolicity that extends properly the sectional-hyperbolicity and includes the Rovella attractor as a archetypal example. The main feature of this definition is the existence of arbitrarily large hyperbolic times for points outside the stable manifolds of the singularities. In this paper we will prove that any attractor associated to a $C^1$ vector field $X$ on a three-dimensional manifold satisfying this kind of hyperbolicity is rescaling-expansive and presents sensitiveness respect to initial conditions.