7 problems

Dense-graph synchronization conjecture. There exist graphs with minimum degree approaching $0.75n$ that do not synchronize.

Persistence conjecture. Similar persistence results should hold for more complex bifurcation scenarios that occur only in infinite-dimensional systems.

Consider the stochastic nearest-neighbor Kuramoto model with interaction range $r>1$, system size $n$, twisted states indexed by $q$, prefactor $C(q,n)$, and energy barrier…

Let $\mu$ be a symmetric unimodal distribution of natural frequencies, and let $V_{1}^{\mu}(r_{1},r_{2})$ and $V_{2}^{\mu}(r_{1},r_{2})$ denote the two-community self-consistency m…

Let $\mu$ be the common distribution of natural frequencies in the two communities, let $\psi=\psi_{2}-\psi_{1}\in\{0,\pi\}$, and let $r_{1},r_{2}$ be the two synchronization order…

Let $\theta$ denote the phase on the circle, let $\theta_{1}$ and $\theta_{2}$ be the average phases of the two communities, and write their phase difference as … In the disorder-f…

Xie–Kelley–Leonard conjecture. The maximum number of equilibria is $10$, strictly smaller than the upper bound $14$.