The modulation and control of oscillatory phenomena in complex systems are essential with implications across biological, physical, neural, and engineered systems. Beyond pairwise interactions, higher-order networks have emerged as a valid framework for modeling complex systems, capturing interactions that simultaneously occur among three or more units. However, the influence of higher-order interactions on amplitude death (AD) behavior in complex networks remains poorly understood. Here, we investigate the phenomenon of AD in complex networks featuring delayed coupling across first-order, second-order, and combined interaction schemes. Employing a dimensionality reduction approach, we simplify the high-dimensional system and analytically derive the boundaries of the AD region from the resulting low-dimensional system. Numerical simulations are performed to validate the theoretical framework. Our results indicate that both the coupling strengths of the first-order and second-order interactions, as well as the network topologies, can significantly modulate the oscillatory behavior. The results demonstrate that an increase in either first-order or second-order strength reduces the AD region and promotes two distinct transition processes: a sequential transition from oscillation to AD and back to oscillation or a direct transition from the AD state to oscillatory dynamics. Furthermore, increased connection density enhances oscillatory dynamics, while sparser connectivity promotes the occurrence of the AD state. These findings offer insights into how higher-order interactions shape oscillatory dynamics in complex network architectures.

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• Collective behavior in networks
• Coupled oscillators