An alternate mechanism for the emergence of localized oscillatory patterns in networked reaction–diffusion systems

Localized oscillatory patterns, confined to finite spatial domains or subsets of nodes in a network, have been observed in cortical neurons, metacommunity ecosystems, and chemical experiments. Previous studies have shown that wave bifurcations (oscillatory Turing instabilities) in undirected networked reaction–diffusion systems can induce such localized oscillations. Here, we propose an alternative mechanism for their emergence, namely Hopf-type instabilities acting on a subcritical Turing branch. Unlike wave bifurcations, this mechanism does not require interactions amongst at least three components. We further demonstrate its robustness in both the Brusselator and FitzHugh-Nagumo models in undirected Barabási–Albert, Erdős–Rényi, and Watts–Strogatz networks. This work extends the universality of such patterns in networked systems and provides a theoretical foundation for the control of collective dynamical phenomena in complex systems.