Lyapunov functional existence conjecture for the inertial and first-order Kuramoto models

Let $\mathcal{V}=(\nu_1,\ldots,\nu_N)$ be the natural-frequency vector, let $\mathcal{D}(\mathcal{V})$ denote its frequency-diameter quantity, and let $\kappa_c(\mathcal{V})$ be the critical coupling strength above which phase-locked states exist. A weak Lyapunov functional is a Lyapunov functional in the weak sense used for the systems. Lyapunov functional existence conjecture. (Weak form) There is a constant $c\geq\frac12$ such that if

$$\kappa>c\mathcal{D}(\mathcal{V}),$$

then both the inertial system $$ and the first-order system $$ admit weak Lyapunov functionals. (Strong form) If

$$\kappa>\kappa_c(\mathcal{V}),$$

then both $$ and $$ admit weak Lyapunov functionals. Constructing such a functional is proposed as an inertia-independent route toward resolving complete synchronization for all inertial regimes; the supplied text explicitly describes this as an open problem.