The Kuramoto model of coupled oscillators with a bi-harmonic coupling function

• The Kuramoto model with a bi-harmonic coupling function was investigated.
• We develop a method for an analytic solution of self-consistent equations.
• We observed a multi-branch locking with a multiplicity of coherent states.
• Multi-branch synchronous states coexist with neutrally stable asynchronous regime.
• We show that the asynchronous state has a finite life time for finite ensembles.

We study synchronization in a Kuramoto model of globally coupled phase oscillators with a bi-harmonic coupling function, in the thermodynamic limit of large populations. We develop a method for an analytic solution of self-consistent equations describing uniformly rotating complex order parameters, both for single-branch (one possible state of locked oscillators) and multi-branch (two possible values of locked phases) entrainment. We show that synchronous states coexist with the neutrally linearly stable asynchronous regime. The latter has a finite life time for finite ensembles, this time grows with the ensemble size as a power law.