Scaling and synchronization in a ring of diffusively coupled nonlinear oscillators

TL;DR

This study investigates chaos synchronization in a ring of coupled nonlinear oscillators, demonstrating how additional couplings and external drive influence the maximum synchronized oscillators, with findings supported by numerical simulations and Lyapunov analysis.

Contribution

It introduces a method to increase the number of synchronized oscillators beyond size instability limits using additional couplings and explores the scaling relations with critical coupling strength.

Findings

Synchronization can be enhanced with additional couplings.

An exponential relation exists between synchronized oscillators and coupling strength.

Synchronization error decays as a power law with noise intensity.

Abstract

Chaos synchronization in a ring of diffusively coupled nonlinear oscillators driven by an external identical oscillator is studied. Based on numerical simulations we show that by introducing additional couplings at $(mN_c+1)$-th oscillators in the ring, where $m$ is an integer and $N_c$ is the maximum number of synchronized oscillators in the ring with a single coupling, the maximum number of oscillators that can be synchronized can be increased considerably beyond the limit restricted by size instability. We also demonstrate that there exists an exponential relation between the number of oscillators that can support stable synchronization in the ring with the external drive and the critical coupling strength $\epsilon_c$ with a scaling exponent $\gamma$. The critical coupling strength is calculated by numerically estimating the synchronization error and is also confirmed from the conditional Lyapunov exponents (CLEs) of the coupled systems. We find that the same scaling relation exists for $m$ couplings between the drive and the ring. Further, we have examined the robustness of the synchronous states against Gaussian white noise and found that the synchronization error exhibits a power-law decay as a function of the noise intensity indicating the existence of both noise-enhanced and noise-induced synchronizations depending on the value of the coupling strength $\epsilon$. In addition, we have found that $\epsilon_c$ shows an exponential decay as a function of the number of additional couplings. These results are demonstrated using the paradigmatic models of R\"ossler and Lorenz oscillators.