A Stability Criterion for Relational Dynamics: Kuramoto as a Canonical Test Case

This paper proposes a general stability (or restorability) criterion for relational dynamical systems, formulated independently of synchronization or order parameters. While synchronization phenomena in coupled oscillators have traditionally been analyzed through phase coherence and macroscopic order parameters, such approaches often conflate observable collective behavior with the underlying notion of structural stability. The central contribution of this work is a relational stability criterion that characterizes stability not by convergence to a fixed point or synchronized state, but by the existence of a restorative relational flow. Stability is defined by the ability of a system to return perturbed configurations to the same relational equivalence class, as quantified through relational observables and a Lyapunov-type functional governing their evolution. The Kuramoto model is analyzed as a canonical test case in which the proposed criterion is realized in an explicit and physically transparent manner. Rather than treating Kuramoto dynamics as a special synchronization mechanism, the analysis shows that it provides a minimal and fully visible instance of relational stability, with synchronization emerging as a particular manifestation rather than a defining principle. The relational formulation further clarifies the long-standing degeneracy observed in low-dimensional systems. In particular, the pathological behavior of the N=2N=2N=2 Kuramoto system is shown to arise from a geometric collapse of relational space, rather than from mean-field assumptions or coupling details. Although relational coordinates remain definable, structural stability itself becomes ill-posed. By decoupling stability from synchronization, the proposed framework provides an operational lens applicable beyond the Kuramoto setting, including biological, neural, and socio-dynamical systems in which conventional order parameters are ill-defined or misleading. The results suggest that stability is best understood as a structural grammar of relational dynamics, with synchronization representing only one of its possible expressions.