Synchronization in repulsively coupled oscillators

A long-standing expectation is that two repulsively coupled oscillators tend to oscillate in opposite directions. It has been difficult to achieve complete synchrony in coupled identical oscillators with purely repulsive coupling. Here, we introduce a general coupling condition based on the linear matrix of dynamical systems for the emergence of the complete synchronization in pure repulsively coupled oscillators. The proposed coupling profiles (coupling matrices) define a bidirectional cross-coupling link that plays the role of indicator for the onset of complete synchrony between identical oscillators. We illustrate the proposed coupling scheme on several paradigmatic two-coupled chaotic oscillators and validate its effectiveness through the linear stability analysis of the synchronous solution based on the master stability function approach. We further demonstrate that the proposed general condition for the selection of coupling profiles to achieve synchronization even works perfectly for a large ensemble of oscillators.