Asymmetric coherent clusters with microscopic nonstationarity in network of Stuart-Landau oscillators.

We study the emergence of a unique dynamical state, termed the asymmetric coherent cluster (ACC), in ring networks of identical Stuart-Landau oscillators arising from the interplay of symmetric and asymmetric couplings. The ACC state constitutes an intermediate regime between the splay (SP) and complete synchronization (CS) states, exhibiting macroscopic coherence despite persistent microscopic phase drift of the coupled oscillators. Remarkably, in the ACC state, oscillators self-organize into a stationary, symmetry-broken macroscopic structure while remaining dynamically nonstationary at the microscopic level. The transition from the SP state to the ACC state occurs via an abrupt frequency-unlocking process and is mediated by the breakdown of a stable limit cycle into an invariant torus. In contrast, the ACC-CS transition proceeds through gradual frequency locking accompanied by a continuous increase in global coherence. In this regime, the invariant torus collapses smoothly onto a fully synchronized limit cycle via a reverse supercritical Neimark-Sacker bifurcation. Numerical simulations support the theoretical analysis and demonstrate the robustness of the ACC state. These results demonstrate how competing symmetric and asymmetric interactions give rise to nontrivial collective dynamics, offering a route to coherent macroscopic states beyond conventional phase-locked synchronization. Furthermore, the robustness of ACCs and their existence over a broad range of coupling parameters strongly suggest their experimental realizability.