Dynamics of Kuramoto oscillators with time-delayed positive and negative couplings

Many real-world examples of distributed oscillators involve not only time delays but also attractive (positive) and repulsive (negative) influences in their network interactions. Here, considering such examples, we generalize the Kuramoto model of globally coupled identical oscillators with time-delayed positive and negative couplings to explore the effects of such couplings in collective phase synchronization. We analytically derive the exact boundaries for stable incoherent and coherent states in terms of the system parameters allowing us to examine the interplay of symmetric and asymmetric time delays and couplings in collective synchronization. Dependent on these parameters, regions of coherent, incoherent, and mixed (coherent, partially coherent, incoherent) states with hysteresis are possible. The region of stability for incoherent states decreases with increasing time delay in all cases and it overall gets reduced in the presence of a time delay in repulsive coupling. The time-delay effects for instability can become more significant at the delay values that are about half an oscillation period length, or its multiples in the case of positive time-delay couplings, and is about a full period or its multiples in case of negative delayed couplings. The mixed state region shows multistability among fully coherent, fully incoherent, and partially coherent (clustered) states. Partially coherent or clustered states occur near the instability-stability boundaries and can show quasiperiodic and nonstationary behaviors. We discuss the implications of the model and the results for natural systems, particularly neuronal network systems in the brain.