Analysis of Bifurcation-Induced Indecisive Limit Cycles in Hybrid Opinion Dynamics with Social Attention-Controlled Consensus.

Shared beliefs in social interactions evolve through opinion exchanges governed by hybrid time scales and heterogeneous update rules. In this letter, we rigorously analyze hybrid opinion dynamics (HOD) of an interacting continuous-discrete dyad. By characterizing the HOD via a stroboscopic map, we analytically establish that high reactivity triggers a local flip bifurcation that destabilizes consensus. Following this loss of stability, the opinion trajectories enter a saturated plateau-switching regime and converge to a stable indecisive limit cycle representing a persistent cyclic deadlock. Leveraging Floquet theory and a Lyapunov-based cycle-to-cycle contraction, we prove the asymptotic stability of these oscillations. Finally, we propose a social-attention-based control law that guarantees a transition from cyclic deadlock to consensus.