# Enhancement of Network Synchronizability via Two Oscillatory System

## TL;DR

This paper demonstrates that using a pair of oscillators per node can enhance network synchronizability at high coupling strengths, reducing the need for vector coupling and improving stability in chaotic and hyperchaotic systems.

## Contribution

It introduces the two oscillatory system (TOS) approach, showing that a single coordinate exchange suffices for synchronization, unlike traditional vector coupling methods.

## Key findings

- TOS improves synchronization stability at large coupling strengths.
- Numerical validation in chaotic and hyperchaotic systems.
- Synchronization threshold increases with TOS, varying by oscillator type.

## Abstract

The loss of synchronizability at large coupling strength is of major concern especially in the fields of secure communication and complex systems. Because theoretically, the coupling mode that can surely stabilize the chaotic/hyperchaotic synchronized state is vector coupling (using all the coordinates) which is in contrast to the practical demand of information exchange using lesser number of coordinates (commonly via a single coordinate). In the present work, we propose that if the node dynamics are given by a pair of oscillators (say, {\it two oscillatory system} TOS) rather than by a conventional way of single oscillator (say, {\it single oscillatory system} SOS), then the information exchange via a single coordinate could be sufficient to stabilize the chaotic/hyperchaotic synchronization manifold at large coupling strength. The frameworks of drive-response system and Master Stability Function (MSF) have been used to study the TOS effect by varying TOS parameters with and without feedback (feedback means quorum sensing conditions). The TOS effect has been found numerically both in the chaotic (R{\"o}ssler, Chua and Lorenz) and hyperchaotic (electrical circuit) systems. However, since threshold also increases as a side effect of TOS, the extent of $\beta$ enhancement depends on the choice of oscillator model like larger for R{\"o}ssler, intermediate for Chua and smaller for Lorenz.