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We study the collective dynamics of identical phase oscillators on globallycoupled networks whose interactions are asymmetric and mediated by positive andnegative couplings. We split the set of oscillators into two interconnectedsubpopulations. In this setup, oscillators belonging to the same group interactvia symmetric couplings while the interaction between subpopulations occurs inan asymmetric fashion. By employing the dimensional reduction scheme of theOtt-Antonsen (OA) theory, we verify the existence of traveling-wave and$\pi$-states, in addition to the classical fully synchronized and incoherentstates. Bistability between all collective states is reported. Analyticalresults are generally in excellent agreement with simulations; for someparameters and initial conditions, however, we numerically detect chimera-likestates which are not captured by the OA theory.

Discordant synchronization patterns on directed networks of identical phase oscillators with attractive and repulsive couplings