Higher-order synchronization on the sphere

Abstract

We construct a system of $N$ interacting particles on the unit sphere $S^{d-1}$ in $d$-dimensional space, which has $d$-body interactions only. The equations have a gradient formulation derived from a rotationally-invariant potential of a determinantal form summed over all nodes, with antisymmetric coefficients. For $d=3$, for example, all trajectories lie on the $2$-sphere and the potential is constructed from the triple scalar product summed over all oriented $2$-simplices. We investigate the cases $d=3,4,5$ in detail, and find that the system synchronizes from generic initial values, for both positive and negative coupling coefficients, to a static final configuration in which the particles lie equally spaced on $S^{d-1}$. Completely synchronized configurations also exist, but are unstable under the $d$-body interactions. We compare the relative effect of $2$-body and $d$-body forces by adding the well-studied $2$-body interactions to the potential, and find that higher-order interactions enhance the synchronization of the system, specifically, synchronization to a final configuration consisting of equally spaced particles occurs for all $d$-body and $2$-body coupling constants of any sign, unless the attractive $2$-body forces are sufficiently strong relative to the $d$-body forces. In this case the system completely synchronizes as the $2$-body coupling constant increases through a positive critical value, with either a continuous transition for $d=3$, or discontinuously for $d=5$. Synchronization also occurs if the nodes have distributed natural frequencies of oscillation, provided that the frequencies are not too large in amplitude, even in the presence of repulsive 2-body interactions which by themselves would result in asynchronous behaviour.

Multisoliton complex systems with explicit superpotential interactions

“…The asymptotic values of the nodes are therefore equally spaced on the half-circle, a configuration which is sometimes referred to as a splay state, see the discussion of properties of splay states and further references in [60]. The fact that the states θ i = θ 0 ± i π N are fixed points is proved in [59], section 7.1, using the property…”

“…These equations have been previously studied [59] (section 7), as well as a similar system with symmetric all-to-all coupling [54] (section 4). The 2π-periodic potential is defined by (2) with the zeroes, which are absolute minima, given by W θi = 0 for all i.…”

Higher-order connection Laplacians for directed simplicial complexes

“…By formulating the higher-order Connection Laplacians this work demonstrates that a much unexplored yet very promising research direction in higher-order networks is the investigation of the dynamics of topological signals combined with rich algebraic structures. Other examples of this emerging topic are the adoption of the Dirac operator [66] and its associated gamma matrices [67] to study for instance Turing and Dirac-induced patterns of topological signals, higher-order synchronization dynamics [68,69], the use of sheaves in opinion dynamics [70], and the extensive literature on physics-inspired neural networks [43,71]. The mathematical methods that we present may also be used to develop random-walk centrality measures [72] and embeddings [55] for directed higher-order networks.…”