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Synchronization in bipartite networks of oscillators with coupling delays
Nirmal Punetha, Ramakrishna Ramaswamy and Fatihcan M. Atay
We study bipartite networks of phase oscillators with general nonlinear coupling and distributed time delays. In particular, we consider phase-locked synchronous solutions where the oscillators in each partition are perfectly synchronized with each other but have a phase difference with those in the other partition. We show that the phase difference must be either zero or $\pi$ radians and derive analytical conditions for the stability of both types of solutions. The stability condition implies that the network can have several co-existing stable solutions. In fact, the number of stable in-phase and anti-phase phase-locked solutions with different collective frequencies grows without bound with increasing delay, and the system exhibits multistability, hysteresis, phase flips, and sensitivity to disturbances. Finally, we apply our results to networks of Landau-Stuart and Rössler oscillators and show that the theory successfully predicts in-phase and anti-phase synchronous behavior in appropriate parameter ranges.