Riddling and chaotic synchronization of coupled piecewise-linear Lorenz maps

TL;DR

This paper studies how coupled piecewise-linear Lorenz maps exhibit riddled basins of attraction, affecting predictability, and provides rigorous proof of their existence in certain parameter ranges.

Contribution

It demonstrates the occurrence of riddled basins in coupled Lorenz maps and offers a rigorous mathematical framework for their analysis.

Findings

Riddled basins occur over wide parameter ranges.

Riddled basins impact the system's predictability.

Mathematical conditions for riddled basins are established.

Abstract

We investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise-linear Lorenz maps. Riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale, which has serious consequences on the predictability of the final state for such a coupled system. We found that there are wide parameter intervals for which two piecewise-linear Lorenz maps exhibit riddled basins (globally or locally), which indicates that there are riddled basins in coupled Lorenz equations, as previously suggested by numerical experiments. The use of piecewise-linear maps makes it possible to prove rigorously the mathematical requirements for the existence of riddled basins.