Attracting Rotations Obstruct Almost-Global Consensus on Spheres over Strongly Connected Graphs

Abstract

We study discrete-time consensus on the Euclidean unit sphere, where weighted combinations of neighboring states are normalized to unit length. For symmetric weights on connected undirected graphs and ambient dimension $d\ge3$, earlier results establish instability of non-consensus fixed points under suitable weight conditions. For directed graphs on the unit circle, earlier results show that fixed points other than consensus and antipodal configurations occur only for a measure-zero set of weights. We show that these conclusions do not extend directly to general strongly connected directed graphs in higher dimensions. First, we construct full-rank non-consensus fixed points whose tangent eigenvalues all equal one, showing that strict linear instability need not hold. In a separate construction, for every $d\ge3$, we present a sparse strongly connected graph admitting full-rank fixed points on an open set of its permitted weights. Our main result establishes robust counterexamples to almost-global consensus: attracting rotating motions exist for $d=2$ with every $n\ge3$ and for $d=3$ with every $n\ge6$. These motions are locally asymptotically stable modulo common rotations and have asymptotic phase. They persist on open sets of strictly positive weight matrices. For each such matrix, an open set of initial configurations approaches neither consensus nor any fixed configuration. Thus the absence of full-rank non-consensus fixed points does not ensure almost-global convergence to consensus.