Solvable collective dynamics of globally coupled Stuart-Landau limit-cycle systems under mean-field feedback

Abstract

Coupled Stuart-Landau limit-cycle system serves as an important paradigmatic model for studying synchronization transitions and collective dynamics in self-sustained nonlinear systems with amplitude degree of freedom. In this paper, we extensively investigate three typical solvable collective behaviors in globally coupled Stuart-Landau limit-cycle systems under mean-field feedback: incoherence, amplitude death, and locked states. In the thermodynamic limit of \begin{document}$N\rightarrow\infty$\end{document}, the critical condition characterizing the transition from incoherence to synchronization is explicitly obtained via performing the linear stability of the incoherent states, it is found that the synchronization transition occurs at a smaller coupling strength when the strength of mean-field feedback is gradually enhanced; the stable regions of amplitude death are theoretically obtained via an analysis of the linear stability of coupled systems around the origin, it is unveiled that the presence of mean-field feedback is able to effectively eliminate the phenomenon of amplitude death in the coupled systems; furthermore, the existence of locked states is theoretically analyzed, and in particular the boundary of stable amplitude death region is re-derived from the self-consistent relation of the order parameter for the locked states. The study of this work reveals the key role of mean-field feedback in controlling the collective dynamics of coupled nonlinear systems, deepens the understanding of the impact of mean-field feedback technique on the coupling-induced collective behaviors, and is beneficial for us to further understand the emergence rules and the underlying mechanisms of self-organized behaviors in complex coupled systems.