A Geometric Obstruction to Almost Global Synchronization on Riemannian Manifolds

Multi-agent systems on nonlinear spaces sometimes fail to synchronize. This is usually attributed to the initial configuration of the agents being too spread out, the graph topology having certain undesired symmetries, or both. Besides nonlinearity, the role played by the geometry and topology of the nonlinear space is often overlooked. This paper concerns two gradient descent flows of quadratic disagreement functions on general Riemannian manifolds. One system is intrinsic while the other is extrinsic. We derive necessary conditions for the agents to synchronize from almost all initial conditions when the graph used to model the network is connected. If a Riemannian manifold contains a closed curve of locally minimum length, then there is a connected graph and a dense set of initial conditions from which the intrinsic system fails to synchronize. The extrinsic system fails to synchronize if the manifold is multiply connected. The extrinsic system appears in the Kuramoto model on $\smash{\mathcal{S}^1}$, rigid-body attitude synchronization on $\mathsf{SO}(3)$, the Lohe model of quantum synchronization on the $n$-sphere, and the Lohe model on $\mathsf{U}(n)$. Except for the Lohe model on the $n$-sphere where $n\in\mathbb{N}\backslash\{1\}$, there are dense sets of initial conditions on which these systems fail to synchronize. The reason for this difference is that the $n$-sphere is simply connected for all $n\in\mathbb{N}\backslash\{1\}$ whereas the other manifolds are multiply connected

High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally

The Kuramoto model of a system of coupled phase oscillators describe synchronization phenomena in nature. We propose a generalization of the Kuramoto model where each oscillator state lives on the compact, real Stiefel manifold St(p,n). Previous work on high-dimensional Kuramoto models have largely been influenced by results and techniques that pertain to the original model. This paper uses optimization and control theory to prove that the generalized Kuramoto model on St(p,n) converges to a completely synchronized state for any connected graph from almost all initial conditions provided (p,n) satisfies p<=2n/3-1 and all oscillator frequencies are equal. This result could not have been predicted based on knowledge of the Kuramoto model in complex networks on the circle with homogeneous oscillator frequencies. In that case, almost global synchronization is graph dependent; it applies if the network is acyclic or sufficiently dense. The problem of characterizing all such graphs is still open. This paper hence identifies a property that distinguishes many high-dimensional generalizations of the Kuramoto model from the original model. It should therefore have important implications for modeling of synchronization phenomena in physics and control of multi-agent systems in engineering applications

From the Lohe tensor model to the Lohe Hermitian sphere model and emergent dynamics

We study emergent behaviors of the Lohe hermitian sphere(LHS) model which is an aggregation model on ${\mathbb C}^d$. The LHS model is a complex analog of the Lohe sphere model on ${\mathbb R}^d$, and hermitian spheres are invariant sets for the LHS dynamics. For the derivation of the LHS model, we use a top-down approach, namely a reduction from a high-rank aggregation model, "$\textit{the Lohe tensor model}$." The Lohe tensor model is a first-order aggregation model on the space of tensors with the same rank and sizes, and it was first proposed by the authors in a recent work \cite{H-P}. In this work, we study how the LHS model appears as a special case of the Lohe tensor model and for the proposed model, we provide a cross-ratio like conserved quantity, a sufficient framework for the complete aggregation and a uniform $\ell^p$-stability estimate with respect to initial data