In-phase synchronization in complex oscillator networks by adaptive delayed feedback control

TL;DR

This paper presents an adaptive delayed feedback control method to achieve in-phase synchronization in complex oscillator networks, even below the natural synchronization threshold, with analytical and numerical validation.

Contribution

It introduces a practical adaptive control algorithm using time-delay adjustments to synchronize nearly identical oscillators in networks, regardless of coupling strength.

Findings

The control method achieves in-phase synchronization below the coupling threshold.

Adaptive delay adjustment minimizes control power while maintaining synchronization.

Numerical simulations confirm the effectiveness of the proposed approach.

Abstract

In-phase synchronization is a special case of synchronous behavior when coupled oscillators have the same phases for any time moments. Such behavior appears naturally for nearly identical coupled limit-cycle oscillators when the coupling strength is greatly above the synchronization threshold. We investigate the general class of nearly identical complex oscillators connected into network in a context of a phase reduction approach. By treating each oscillator as a black-box possessing a single-input single-output, we provide a practical and simply realizable control algorithm to attain the in-phase synchrony of the network. For a general diffusive-type coupling law and any value of a coupling strength (even greatly below the synchronization threshold) the delayed feedback control with a specially adjusted time-delays can provide in-phase synchronization. Such adjustment of the delay times performed in an automatic fashion by the use of an adaptive version of the delayed feedback algorithm when time-delays become time-dependent slowly varying control parameters. Analytical results show that there are many arrangements of the time-delays for the in-phase synchronization, therefore we supplement the algorithm by an additional requirement to choose appropriate set of the time-delays, which minimize power of a control force. Performed numerical validations of the predictions highlights the usefulness of our approach.