# Statistical instability for contracting Lorenz flows

## TL;DR

This paper demonstrates that within certain parameter families of modified Lorenz flows, the statistical properties are unstable, meaning small changes in parameters can lead to significant differences in long-term behavior.

## Contribution

It introduces a new class of Lorenz flows with contracting eigenvalues and proves their statistical instability, extending understanding of Lorenz attractors under different eigenvalue conditions.

## Key findings

- No statistical stability for the parameter set with physical measures
- Instability shown at the level of contracting Lorenz maps
- Results extend to modified geometric Lorenz attractors

## Abstract

We consider one parameter families of vector fields introduced by Rovella, obtained through modifying the eigenvalues of the geometric Lorenz attractor, replacing the expanding condition on the eigenvalues of the singularity by a contracting one. We show that there is no statistical stability within the set of parameters for which there is a physical measure supported on the attractor. This is achieved obtaining a similar conclusion at the level of the corresponding one-dimensional contracting Lorenz maps.