11 problems

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 th…

Let $u1,\ldots, uN$ be fixed natural frequencies, and let $u=( u1,\ldots, uN)$. Consider the inertial Kuramoto system $$, with coupling strength$kappa$, phase initial data$Theta…

Generic-data phase-locking conjecture. (Weak form) For every $\varepsilon>0$, there exists $\delta=\delta(\varepsilon)$ such that, if $\mathcal{D}(\mathcal{V})/\kappa<\delta$, then…

Weak-Lyapunov-functional conjecture. Either there is a constant $c\ge\frac12$ such that, whenever $\kappa>c\mathcal{D}(\mathcal{V})$, both models admit weak Lyapunov functionals, o…

Let the dynamic Kuramoto model be given by with zero natural frequencies, and let the phase vector start from a deterministic point outside the set of spurious fixed points…

Consider the finite-particle all-to-all Kuramoto model with natural-frequency vector $\Omega$ and frequency diameter $D(\Omega)$. Let $\Theta^0$ be an initial phase vector, and let…

Let $N$ be the number of particles, let $\Theta^0$ be an initial phase vector, and let $\Omega=(\nu_1,\ldots,\nu_N)$ be the vector of natural frequencies. Write $D(\Omega)$ for the…

Let $W^u(\frac{1}{2\pi})$ be the unstable manifold of the incoherent equilibrium for the diffusive Kuramoto equation, and let $\mathcal M_{0A}$ be the corresponding invariant manif…

Let $\Phi_\mathcal H$ be the graph parametrizing the two-dimensional global attractor $W^u(\frac{1}{2\pi})$, let $q_{OA}(z)$ denote the corresponding state on the Ott–Antonsen mani…

Let $q_h$ be the heteroclinic solution of the Kuramoto equation connecting the incoherent equilibrium to the synchronized equilibrium, and let $W^{\frac{1}{2\pi}}$ denote the globa…

Let $N$ phase oscillators evolve according to the homogeneous Kuramoto system … where $\theta_0\in\mathbb{R}^N$ and $R(\theta)$ is the magnitude of the order parameter. Global phas…