# Robust Chaos and the Continuity of Attractors

## TL;DR

This paper investigates the continuity of chaotic attractors in parameterized maps, proposing that for piecewise-smooth maps, this concept helps identify robust chaotic regions, supported by theoretical conditions and examples like skew tent maps and the Lozi map.

## Contribution

It introduces conditions for the continuity of attractors in piecewise-smooth maps and demonstrates how this concept delineates robust chaos regions, unlike in smooth unimodal maps.

## Key findings

- Continuity of attractors can be established for certain piecewise-smooth maps.
- Robust chaos regions can be characterized using attractor continuity.
- Examples include coupled skew tent maps and the Lozi map.

## Abstract

As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely, but that for piecewise-smooth maps it provides a way to delineate structure within parameter regions of robust chaos and form a stronger notion of robustness. We obtain conditions for the continuity of an attractor and demonstrate the results with coupled skew tent maps, the Lozi map, and the border-collision normal form.