On the synchronization of ReLU version of dense networks
Journal
Chaos, Solitons & Fractals
Journal Volume
202
Start Page
117444
ISSN
0960-0779
Date Issued
2026-01
Author(s)
Shiu, Ruey-An
Abstract
In this article, we analyze a Kuramoto-type system in which the classical sine coupling is replaced by the Rectified Linear Unit (ReLU) function max { 0 , sin ( x ) } . Both identical and non-identical oscillators are studied on connected graphs, and non-identical oscillators are also examined on two-group social networks. For the identical oscillators, our results show that simple connectivity of the network suffices for synchronization in both the classical and ReLU-modified models. For the non-identical oscillators, since the connection between oscillators is not all-to-all and the coupling function is the ReLU sine function, this causes the complexity of the synchronization analysis. To overcome this subtlety, we design a special Ansatz to proceed the proof by contradiction. In such a setting, by assuming the oscillators will not achieve a frequency synchronization asymptotically and analyzing the integrability of the difference of the second derivatives of phase functions for carefully chosen N pairs of connected oscillators, we show there should be at least N + 1 distinct oscillators in this ReLU system, which violates the fact that the total number of the oscillators is N . Moreover, we show that the synchronized frequency for the ReLU model is the maximum among all the natural frequencies; while it is well-known that the synchronized frequency in the classical Kuramoto model is the mean of all the natural frequencies. In the social-network setting, characterized by intra-group competition and inter-group attraction, we derive explicit conditions that guarantee asymptotic synchronization. Numerical experiments support the theory and show that the classical model converges faster on equivalent topologies. Overall, our results provide a theoretical foundation for understanding synchronization in systems with asymmetric competitive–cooperative interactions.
Publisher
Elsevier BV
Type
journal article