Asymptotic behavior of a second-order swarm sphere model and its kinetic limit

Abstract

We study the asymptotic behavior of a second-order swarm model on the unit sphere in both particle and kinetic regimes for the identical cases. For the emergent behaviors of the particle model, we show that a solution to the particle system with identical oscillators always converge to the equilibrium by employing the gradient-like flow approach. Moreover, we establish the uniformin-time 2-stability using the complete aggregation estimate. By applying such uniform stability result, we can perform a rigorous mean-field limit, which is valid for all time, to derive the Vlasov-type kinetic equation on the phase space. For the proposed kinetic equation, we present the global existence of measure-valued solutions and emergent behaviors.

Synchronization of a Generalized High-dimensional Kuramoto Model With Directed Graphs

“…It follows that r(0) belongs to S ε v denoted by (20) in Lemma 2. Thus, by the invariance of S ε v , we have…”

“…In [6], the case of heterogeneous frequencies is investigated. In [20,26], the second-order high-dimensional Kuramoto model is discussed. In [12], based on the global order parameter, the d-dimensional Kuramoto model with positive feedback is considered.…”

Robust synchronization of heterogeneous robot swarms on the sphere

“…The study of this model in physics is motivated by its relation to synchronization of quantum bits [8], [9], but it also appears in control systems designed to achieve reduced rigid-body attitude synchronization, i.e., to coordinate the pointing orientations of robots [6], [22], [23], in bio-inspired models of source-seeking and learning [12], [24], and in machine learning applications [13]. There are several variations of the model, including second-order dynamics [25], [26] and discrete-time maps [27]. A limitation in our current understanding of the Lohe model is the restriction to combinations of complete or acyclic networks [11], [14], [15], [23], [26], homogeneous frequencies [6], [10], [21], [25], and local behavior [10], [11], [25], [26], [28].…”

“…There are several variations of the model, including second-order dynamics [25], [26] and discrete-time maps [27]. A limitation in our current understanding of the Lohe model is the restriction to combinations of complete or acyclic networks [11], [14], [15], [23], [26], homogeneous frequencies [6], [10], [21], [25], and local behavior [10], [11], [25], [26], [28]. By contrast, this paper concerns the global behavior of the Lohe model with heterogeneous frequencies over non-trivial networks.…”