A graphical approach to estimate the critical coupling strength for Kuramoto networks

The Kuramoto model is an archetypal model for studying synchronization in groups of nonidentical oscillators. Each oscillator is imbued with its own personal inherent driving frequency and experiences attractive coupling forces toward all the other oscillators in the system. As the coupling increases, there exists a minimal coupling strength called the critical coupling beyond which the system moves in a collective rhythm. A unified approach for creating approximations of the critical coupling is created. It is based on an interpretation of a measurement of phase synchronization among the oscillators (the order parameter) as a function of the coupling strength. The approach allows a graphical way to develop new approximations that are provably, strict lower bounds. It is shown that several of the critical coupling bounds that have been previously studied can be interpreted in this unified framework. In addition, a process based on fixed point sampling is introduced that converts upper bounds for the critical coupling into associated lower bounds.