Analytical calculation of the transition to complete phase synchronization in coupled oscillators

TL;DR

This paper derives exact analytical solutions for the critical coupling strength leading to complete phase synchronization in a chain of coupled oscillators, considering different boundary conditions and validating with numerical results.

Contribution

It provides the first exact solutions for the transition to full synchronization in coupled oscillators with various boundary conditions.

Findings

Exact critical coupling values for periodic and fixed-end boundary conditions.

Analytical results match numerical simulations.

Insights into how boundary conditions affect synchronization transition.

Abstract

Here we present a system of coupled phase oscillators with nearest neighbors coupling, which we study for different boundary conditions. We concentrate at the transition to total synchronization. We are able to develop exact solutions for the value of the coupling parameter when the system becomes completely synchronized, for the case of periodic boundary conditions as well as for an open chain with fixed ends. We compare the results with those calculated numerically.