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
$N\rightarrow\infty$
, 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. The results indicate that the existence of mean-field feedback can effectively eliminate the amplitude death phenomenon in the coupled systems; furthermore, the existence of locked states is analyzed theoretically, 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. This work reveals the key role of mean-field feedback in controlling the collective dynamics of coupled nonlinear systems, deepens the understanding of the influence of mean-field feedback technology on the coupling-induced collective behaviors, and is conductive to our further understanding of the emerging rules and the underlying mechanisms of self-organized behavior in complex coupled systems.
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