Diffusively coupled heterogeneous oscillators in large networks exhibit complex transitions between ordered and disordered states. Understanding the mechanisms governing the onset of disorder and the emergence of order in such systems remains a central challenge in nonlinear dynamics, statistical physics, and control theory. Here we investigate the collective dynamics of three ensembles of complex Ginzburg-Landau-type oscillators—antirotating, counterrotating, and uniformly rotating—coupled diffusively on a directed ring network. As the coupling strength increases, the system initially exhibits enhanced disorder before undergoing a discontinuous transition to a frequency-locked ordered state, accompanied by hysteresis upon reversing the coupling sweep. Disorder is quantified by the root-mean-square deviations of the frequencies and amplitudes, while ensemble-averaging theory provides analytical predictions for the mean dynamics that are in excellent agreement with numerical simulations. Further, we find that finite-size effects become increasingly pronounced as the network size increases, with the hysteresis width exhibiting a nonlinear, quadratic dependence on $N$. These results provide insight into the self-organization of heterogeneous, nonisochronous limit-cycle oscillators under unidirectional diffusive coupling in ring networks.

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• Bifurcations
• Synchronization transition
• Coupled oscillators