45 problems

Let $G\sim G(n,p)$ be a binomial Erdős–Rényi random graph, where each of the $\binom{n}{2}$ possible edges is present independently with probability $p$. A graph is globally synchr…

Concavity conjecture. The function $F_\kappa$ is concave on $[0,\infty)$ for every $\beta,\sigma,\kappa>0$. The conjecture would imply that the synchronized stationary equilibrium,…

Consider basins of attraction in high-dimensional multistable dynamical systems, including the Kuramoto ring, where basin geometry may consist of a concentrated region near an attr…

Let $n$ identical oscillators evolve on a ring according to the Kuramoto model, and let the $q$-twisted state be the phase-locked state with winding number $q$. Wiley–Strogatz–Girv…

Stationary-solution and stability conjecture. When $m>m_*$, the model should be identical to the case $m<1/2$ in terms of the number of stationary solutions and their stability.

Lower-bound conjecture.

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 $N$ oscillators have natural-frequency vector $\mathcal{V}=(\nu_1,\ldots,\nu_N)$, coupling strength $\kappa$, phase vector $\Theta^0$ as initial data, and order parameter…

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…

Full-characterization conjecture. This third mechanism should be usable to fully characterize the asymptotic behaviors of models and.

Thümmler et al.'s conjecture. Non-trivial phase-locking states exist whenever

Kuramoto's conjecture. The bifurcation diagram of the order parameter $r(t)$ is the one shown in Fig. 2(a).

Let $u_{k,\ell}$ be the continuous twisted state of the heat equation on $\mathbb{T}^d$, for each $k\in\mathbb{Z}$ and $1\leq \ell\leq d$, and let the Kuramoto equation on the rand…

Consider the quaternionic Kuramoto model under the assumptions $\omega_i-\omega_{i+1}=\omega_{i+1}-\omega_{i+2}>0$ and $q_i(0)-q_{i+1}(0)=q_{i+1}(0)-q_{i+2}(0)$ for…

For each interaction parameter $\kappa$, let $v^\gamma$ denote the value function associated with parameter $\gamma\geq 0$, and let $F_\kappa$ be the function defined in the paper…

Let $\kappa_c$ be the critical interaction parameter, and let $\kappa>\kappa_c$. A stationary solution is a probability measure on the torus whose constant flow solves the Kuramoto…

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…

Threshold conjecture. For each fixed $\omega$, there exists a value $\alpha_\omega$ of $\alpha$ such that $\Omega_2$ corresponds to

Consider a network of $n$ identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least $.75(n-1)$ other oscillators. Let $_c$ d…

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…

0.75n critical-threshold conjecture. The critical threshold could be $0.75n$.

Stable-to-1-saddle existence conjecture. For every $N\geq5$, there exists an example of a stable fixed point that loses stability to become a 1-saddle.