Emergent behaviors in group ring flocks

Abstract

We present a first-order aggregation model on a group ring, and study its asymptotic dynamics. In a positive coupling strength regime, we show that the flow generated by the proposed model tends to an equilibrium manifold asymptotically. For this, we introduce a Lyapunov functional which is non-increasing along the flow, and using the temporal decay of the nonlinear functional and the LaSalle invariance principle, we show that the flow converges toward an equilibrium manifold asymptotically. We also show that the structure of an equilibrium manifold is strongly dependent on the structure of an underlying group.

References in corpus (6)

• Collective motion
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• The Spectrum of the Partially Locked State for the Kuramoto Model
• Global phase-locking in finite populations of phase-coupled oscillators
• Synchronization and Stability for Quantum Kuramoto

Cited by in corpus (5)

• Constants of motion for the finite-dimensional Lohe type models with frustration and applications to emergent dynamics
• An intrinsic aggregation model on the special orthogonal group $\mathrm{SO}(3)$: well-posedness and collective behaviours
• Complete aggregation of the Lohe tensor model with the same free flow
• On the Schrödinger-Lohe hierarchy for aggregation and its emergent dynamics
• A gradient flow formulation of the Lohe matrix model with a high-order polynomial coupling