Abstract
Collective motion of self-propelled agents has attracted much attention in vast disciplines. However, almost all investigations focus on such agents evolving in the Euclidean space, with rare concern of swarms on non-Euclidean manifolds. Here we present a novel and fundamental framework for agents evolving on a sphere manifold, with which a variety of concrete cooperative-rules of agents can be designed separately and integrated easily into the framework, which may perhaps pave a way for considering general spherical collective motion (SCM) of a swarm. As an example, one concrete cooperative-rule, i.e., the spherical direction-alignment (SDA), is provided, which corresponds to the usual and popular direction-alignment rule in the Euclidean space. The SCM of the agents with the SDA has many unique statistical properties and phase-transitions that are unexpected in the counterpart models evolving in the Euclidean space, which unveils that the topology of the sphere has an important impact on swarming emergence.
Highlights
CM of agents on a sphere, which itself is an interesting topic, has important implications in analyzing many types of self-propelled agents or continuum-flows evolving on a sphere
As an important example of the generic cooperative-rule (GCR), the spherical direction-alignment (SDA) is provided, which corresponds to the Euclidean direction-alignment (EDA)[22] of agents that is fundamental for the ECM
The spherical collective motion (SCM) with the SDA has many unique characteristics that are unexpectedly distinct from the counterpart ECM model with the EDA, which unveil that the topology of the sphere has an important impact on swarming emergence
Summary
Collective motion of self-propelled agents has attracted much attention in vast disciplines. We present a novel and fundamental framework for agents evolving on a sphere manifold, with which a variety of concrete cooperative-rules of agents can be designed separately and integrated into the framework, which may perhaps pave a way for considering general spherical collective motion (SCM) of a swarm. Compared with the ECM, formulation and characterization of the spherical collective motion (SCM) are much different and complex, since the physical topologies of the sphere and the Euclidean space are distinct. The physical meaning of ζi(k + 1) can be viewed as a certain form of the local polarization surrounding agent i (according to a concrete form of function fg) when all agents are just approaching to their positions at step k + 1 (refer to the notion spherical approaching-direction). One may apply the periodic boundary condition or other boundary conditions to a designated region of interest on the sphere and investigate the agents evolving on that region instead of the whole sphere, this is out of the scope of this paper
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