Binary mixtures of locally coupled mobile oscillators

Synchronized behavior in a system of coupled dynamic objects is a fascinating example of an emerged cooperative phenomena which has been observed in systems as diverse as a group of insects, neural networks, or networks of computers. In many instances, however, the synchronization is undesired because it may lead to system malfunctioning, as in the case of Alzheimer's and Parkinson's diseases, for example. Recent studies of static networks of oscillators have shown that the presence of a small fraction of so-called contrarian oscillators can suppress the undesired network synchronization. On the other hand, it is also known that the mobility of the oscillators can significantly impact their synchronization dynamics. Here, we combine these two ideas-the oscillator mobility and the presence of heterogeneous interactions-and study numerically binary mixtures of phase oscillators performing two-dimensional random walks. Within the framework of a generalized Kuramoto model, we introduce two phase-coupling schemes. The first one is invariant when the types of any two oscillators are swapped, while the second model is not. We demonstrate that the symmetric model does not allow for a complete suppression of the synchronized state. However, it provides means for a robust control of the synchronization timescale by varying the overall number density and the composition of the mixture and the strength of the off-diagonal Kuramoto coupling constant. Instead, the asymmetric model predicts that the coherent state can be eliminated within a subpopulation of normal oscillators and evoked within a subpopulation of the contrarians.