An Analytical Study on the Synchronization of Strange Non-Chaotic Attractors

TL;DR

This paper provides an analytical investigation into the synchronization mechanisms of Strange Non-Chaotic Attractors in coupled quasiperiodically-forced systems, introducing explicit solutions and stability analysis.

Contribution

It presents the first explicit analytical solutions explaining synchronization of SNAs in coupled systems, using bifurcation and stability analysis.

Findings

Synchronization occurs through eigenvalue bifurcation in piecewise linear regions.

Analytical solutions explain the synchronization dynamics.

Master Stability Function confirms the stability of synchronized states.

Abstract

In this paper we present an analytical study on the synchronization dynamics observed in unidirectionally-coupled quasiperiodically-forced systems that exhibit Strange Non-chaotic Attractors (SNA) in their dynamics. The SNA dynamics observed in the uncoupled system is studied analytically through phase portraits and poincare maps. A difference system is obtained by coupling the state equations of similar piecewise linear regions of the drive and response systems. The mechanism of synchronization of the coupled system is realized through the bifurcation of the eigenvalues in one of the piecewise linear regions of the difference system. The analytical solutions obtained for the normalized state equations in each piecewise linear region of the difference system has been used to explain the synchronization dynamics though phase portraits and timeseries analysis. The stability of the synchronized state is confirmed through the Master Stability Function. An explicit analytical solution explaining the synchronization of SNAs is reported in the literature for the first time.