Parametric evolution of unstable dimension variability in coupled piecewise-linear chaotic maps

TL;DR

This paper analytically investigates how unstable dimension variability evolves with parameters in coupled piecewise-linear chaotic maps, providing exact results for its onset and intensity in synchronized systems.

Contribution

It introduces an analytical approach to study the parametric evolution of unstable dimension variability in coupled maps using Markov partitions.

Findings

Exact conditions for the onset of unstable dimension variability.

Quantitative measure of the phenomenon's intensity based on stability of periodic orbits.

Analytical characterization of the phenomenon in synchronized coupled maps.

Abstract

In presence of unstable dimension variability numerical solutions of chaotic systems are valid only for short periods of observation. For this reason, analytical results for systems that exhibit this phenomenon are needed. Aiming to go one step further in obtaining such results, we study the parametric evolution of unstable dimension variability in two coupled bungalow maps. Each of these maps presents intervals of linearity that define Markov partitions, which are recovered for the coupled system in the case of synchronization. Using such partitions we find exact results for the onset of unstable dimension variability and for contrast measure, which quantifies the intensity of the phenomenon in terms of the stability of the periodic orbits embedded in the synchronization subspace.