On the statistical stability of families of attracting sets and the contracting Lorenz attractor

TL;DR

This paper establishes criteria for the statistical stability of attracting sets in vector fields, including Lorenz and Rovella attractors, using dynamical conditions that do not rely on detailed perturbation analysis.

Contribution

It provides verifiable conditions for statistical stability applicable to singular-hyperbolic and contracting Lorenz attractors, including non-robust families like Rovella flows.

Findings

Criteria for statistical stability of attracting sets.

Application to Lorenz and geometrical Lorenz attractors.

Examples of stability in non-robust families of flows.

Abstract

We present criteria for statistical stability of attracting sets for vector fields using dynamical conditions on the corresponding generated flows. These conditions are easily verified for all singular-hyperbolic attracting sets of $C^2$ vector fields using known results, providing robust examples of statistically stable singular attracting sets (encompassing in particular the Lorenz and geometrical Lorenz attractors). These conditions are shown to hold also on the persistent but non-robust family of contracting Lorenz flows (also known as Rovella attractors), providing examples of statistical stability among members of non-open families of dynamical systems. In both instances, our conditions void the use of detailed information about perturbations of the one-dimensional induced dynamics on specially chosen Poincar\'e sections.