84 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…

Let $\mathcal{V}\in\mathbb{R}^N$ be the vector of intrinsic velocities, and let $\kappa_c(\mathcal{V})$ denote the critical coupling strength of the Winfree system. Let initial dat…

Let $N$ oscillators have initial data $(\{\theta_i^0\}_{i=1}^N,\{\omega_i^0\}_{i=1}^N)$ and system parameters $(\{\nu_i\}_{i=1}^N,\kappa,m)$. Write $\mathcal{V}=(\nu_i)_{i=1}^N$ an…

Optimal condition-number threshold conjecture. The optimization landscape of the low-rank factorization problem is benign if

The Lorenz system is perturbed by strong noise, and its top Lyapunov exponent is the largest Lyapunov exponent associated with the resulting stochastic dynamics. Lorenz synchroniza…

Kuramoto's conjecture. There exists a sharp critical coupling

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…

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

For $n>4$, consider the Kuramoto oscillator system on a cycle graph … with periodic boundary conditions. A $q$-twisted state is a phase-locked configuration of the form…

Consider synchronization over $SO(2)$ or over a finite group, with signal $x$ sampled uniformly from group elements, and let $L=L_n$ be the number of frequencies. Multi-frequency s…

A graph is globally synchronizing when its adjacency matrix has only synchronized global minima of the Kuramoto energy. Density-threshold conjecture. For every $\varepsilon>0$, the…

Let $A$ be an $n\times n$ random matrix with zero diagonal and independent off-diagonal entries satisfying … A matrix is globally synchronizing when the only local minima of its Ku…

Let a graph be globally synchronizing when its adjacency matrix is globally synchronizing, meaning that the only local minima of the Kuramoto energy are the global synchronized min…

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

Connectivity obstruction conjecture. With high probability, the only obstruction to synchrony in a random graph is connectivity; equivalently, in the random graph process,

Directed-graph extension conjecture. The proposed feedback law can also apply to the more general case of weighted directed graphs.

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

Synchronization conjecture. The set $\tilde{\mathrm{D}}_{\lambda<0}$ contains the whole of $\mathrm{D}$; equivalently,

Let $G(n,p)$ be an Erdős–Rényi random graph, and consider a system of $n$ identical Kuramoto oscillators on $G(n,p)$. The system globally synchronizes if it converges to a state wi…

Let $\Gamma(t)$ be the global Kuramoto order parameter. An oscillator glass is the proposed glass-like state arising in a system of oscillators with random frustrated interactions…