Synchronized states in a ring of dissipatively coupled harmonic oscillators.

The question under which conditions oscillators with slightly different frequencies synchronize appears in various settings. We consider the case of a finite number of harmonic oscillators arranged on a ring, with bilinear, dissipative nearest-neighbor coupling. We show that by tuning the gain and loss appropriately, stable synchronized dynamics may be achieved. These findings are interpreted using the complex eigenvalues and eigenvectors of the non-Hermitian matrix describing the dynamics of the system. We provide a complete discussion for the case of two oscillators. Ring sizes with a small number of oscillators are discussed taking the case of N=5 oscillators as an example. For N≳10 we focus on the case where the frequency fluctuations of each oscillator are chosen from a Gaussian distribution with zero mean and standard deviation σ. We derive a scaling law for the largest standard deviation σ_{full} that still permits all oscillators to be fully synchronized: σ_{full}∼N^{-3/2}. Finally, we discuss how such random fluctuations influence the timescale on which the synchronized state is reached and on which timescale the synchronized state then decays.