Coherence properties of collective modes in ensembles of oscillators

Synchronization transitions in oscillatory networks often manifest as the emergence of a periodic global mode. While perfect in the thermodynamic limit, this mode fluctuates for finite ensembles. We characterize the coherence of this mode in terms of the phase diffusion constant. In several examples, we always observed normal diffusion, but the dependence of the diffusion constant on the system size <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>D</a:mi> <a:mo>∼</a:mo> <a:msup> <a:mi>N</a:mi> <a:mrow> <a:mo>−</a:mo> <a:mi>μ</a:mi> </a:mrow> </a:msup> </a:mrow> </a:math> depends on the nature of coupled units. For coupled chaotic systems, we found <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>μ</b:mi> <b:mo>=</b:mo> <b:mn>1</b:mn> </b:mrow> </b:math> , while for coupled periodic oscillators we observed, depending on the particular model, values between 2 and 2.7. These large values of the power index are attributed to the size dependence of collective chaos in the finite ensemble, which disappears in the thermodynamic limit. We also show that in the standard Kuramoto model for a symmetric set of frequencies, there is an additional transition to a symmetric chaotic state with vanishing diffusion of the global phase.