Hybrid forcing control of entrainment transitions

Junyi Zheng, Changgui Gu, Jiangsheng Wang and Guolin Wu

Chaos, Solitons & Fractals, 2026, vol. 210, issue P2

Abstract: In many complex systems, external periodic forcing often contains both a constant input and a collectivity-dependent component. Such hybrid forcing appears in contexts ranging from autoregulated neural circuits to dispatch-control schemes in power systems. Most existing studies have either addressed constant forcing and collectivity-dependent forcing separately or left the transition process unexplored. To address this gap, a forced Kuramoto model is established with a tunable hybrid ratio ρ. This parameter continuously interpolates between a constant forcing component and a collectivity-dependent forcing component. The low-dimensional dynamics of the system are derived by means of the Ott–Antonsen reduction. Numerical simulations show that the entrainment transition changes from smooth and reversible to abrupt and hysteretic as ρ exceeds a critical value ρc. Theoretically, this sharp change is attributed to a cusp catastrophe on the self-consistency manifold, which creates bistability and hysteresis. Numerical tests with Gaussian and bimodal frequency distributions further show that the feedback-induced hysteretic entrainment scenario is robust and not limited to the Lorentzian case. This work establishes a unified framework for entrainment-transition classes. It also shows that the trade-off between robustness and sensitivity can be regulated by the hybrid ratio, thereby providing a theoretical basis for entrainment control in neural and engineered networks.

Keywords: Explosive phase transitions; Forced Kuramoto model; Ott–Antonsen reduction; Cusp catastrophe (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:210:y:2026:i:p2:s0960077926008404

DOI: 10.1016/j.chaos.2026.118699

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