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Perceptual graph model systems are designed to be employed when there are discrepancies in decision makers' (DMs) perceptions of a conflict, which may be caused, for instance, by the presence of negative emotion or asymmetric information among DMs. In this case, conventional stability analysis cannot be used; perceptual stability analysis is proposed as a new theoretical procedure that extends existing stability algorithms to situations when DMs have independent perceptions or awarenesses of a conflict. The overriding objective of perceptual stability analysis is to predict possible resolutions, and unveil the dependence of these predictions on variability in a DM's awareness. Perceptual stability analysis takes a two-phase approach. In Phase 1, individual stability analysis is applied to each DM's graph model (a perceptual graph model) from the point of view of the owner of the model, for each DM in the model, using standard or perceptual solution concepts, depending on the owner's awareness of others' perceptions. Then, in Phase 2, metastability analysis is employed to consolidate the stability assessments of a state in all perceptual graph models and across all variants of awareness-i.e., in all possible graph model systems. The distinctive modes of equilibria thus defined reflect the incompatibilities in DMs' perceptions and viewpoints, but nonetheless provide important insights into possible resolutions of the conflict. To demonstrate the practical application of these new developments, a model of the conflict in Chechnya is presented and analyzed.