We investigate a variant of coupled Kuramoto oscillators in which the effective coupling is modulated by the global coherence in a nonlinear fashion. In the thermodynamic limit, a self-consistency analysis reveals the manifestation of coexisting incoherent and coherent states in a finite range of the coupling strength, resulting in the bistability, the width of which is controlled by the feedback strength and the degree of the nonlinearity in the order parameter. For finite populations, intrinsic fluctuations fundamentally reshape this scenario: the synchronization transition manifests even at coupling strengths below the thermodynamic threshold. Strong feedback in combination with weak nonlinearity accelerates the onset of coherence at smaller coupling strengths, whereas strong feedback together with pronounced nonlinearity significantly enlarges the bistable region. Notably, this trend contrasts with the thermodynamic limit, where increasing nonlinearity suppresses the bistability. As the system size increases, the critical coupling strength approaches the thermodynamic-limiting value. Stronger nonlinear feedback facilitates the onset of the coherence closer to the thermodynamic critical coupling even for relatively smaller system sizes, thereby steering both the nature of the transition and its finite-size scaling. In addition, the theoretical estimates of the order parameter using the Ott-Antonsen ansatz and the finite-size scaling also corroborate the observed finite-size effects.

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• Coupled oscillators