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Disorder-to-order transitions and hysteresis in directed ring networks of nonisochronous complex Ginzburg-Landau oscillators

Kumar Sourav1, Naveen Kumar Mendola1, Jung-Wan Ryu2, Awadhesh Prasad3, and Thounaojam Umeshkanta Singh1,*

  • *Contact author: umeshkanta@gmail.com
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Phys. Rev. E 114, 044214 – Published 19 October, 2026

DOI: https://doi.org/10.1103/txbw-g9cd

Abstract

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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Outline

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Phys. Rev. E 114, 044214– Published 19 October, 2026
  • Received 16 February 2026
  • Revised 15 August 2026
  • Accepted 30 September 2026
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