Self-organization on Riemannian manifolds

Figure 1.  Numerical simulation with $ N = 100 $ particles for model (1) in the Euclidean plane with interaction potential given by (13)-(14). (a) Random initial configuration of particles. (b) Starting from the configuration in (a), the model evolves into a uniform particle distribution supported on a disk – see (20). The solid lines represent the trajectories of the particles indicated by stars in figure (a)

Figure 2.  (a) The interaction potential (31) on $ S $ is purely repulsive. The figure indicates how a generic point senses a repelling force from the North pole – see (36). (b) The interaction potential (58) on $ H $ is attractive-repulsive, as the two terms in the right-hand-side of (58) have competing effects. Shown in the figure is a generic point interacting with the vertex of the hyperboloid – see (60)

Figure 3.  Numerical simulation with $ N = 100 $ particles for model (1) on $ S $ with interaction potential given by (31)-(32). (a) Symmetric initial configuration on $ S $, with $ \theta $ coordinates generated randomly in the interval $ \left( \frac{\pi}{8}, \frac{3 \pi}{8} \right) $. (b) The configuration remains symmetric for all times and evolves into a uniform particle distribution supported over the entire sphere – see (42). The solid lines represent the trajectories of the particles indicated by stars in figure (a)

Figure 4.  Numerical simulation with $ N = 100 $ particles for model (1) on $ S $ with interaction potential given by (31)-(32). (a) Random initial configuration on $ S $, with coordinates $ \theta $ and $ \phi $ generated randomly in $ \left( \frac{\pi}{8}, \frac{3 \pi}{8} \right) $ and $ \left(0, \frac{\pi}{2} \right) $, respectively. (b) The configuration in (a) evolves into a uniform particle distribution supported over the entire sphere – see Theorem 3.1. The solid lines represent the trajectories of the particles indicated by stars in figure (a)

Figure 5.  Numerical simulation with $ N = 100 $ particles for model (1) on $ H $ with the interaction potential (58) (see also (56) and (57)). (a) Symmetric (about the vertex) initial configuration on $ H $, with $ \theta $ coordinates generated randomly in the interval $ (0.2,1.25) $. (b) The configuration remains symmetric for all times and evolves into a uniform (with respect to the metric on $ H $) particle distribution supported over a geodesic disk centred at the vertex – see Theorem 4.1. The solid lines represent the trajectories of the particles indicated by stars in figure (a). (c) Zoom-in of figure (b) on the equilibrium configuration

Figure 6.  Numerical simulation with $ N = 100 $ particles for model (1) on $ H $ with the interaction potential (58) (see also (56) and (57)). (a) Random initial configuration on $ H $, with coordinates $ \theta $ and $ \phi $ drawn randomly from intervals $ (0.3,2.3) $ and $ (0,\pi/2) $, respectively. (b) Equilibrium state corresponding to the initial configuration in (a). The solid lines represent the trajectories of the particles indicated by stars. (c) Zoom-in of the equilibrium state in figure (b). Numerical investigations (see Figure 7) suggest that the equilibrium configuration consists of a uniform (with respect to the metric on $ H $) particle distribution supported over a geodesic disk of radius $ R $ (see (71))

Figure 7.  Numerical investigation of the equilibrium configurations on $ H $. (a) Boundary points (filled blue circles) along with the Riemannian centre of mass (red diamond) of the equilibrium state in Figure 6(c). (b) Distances from the Riemannian centre of mass to the particles on the equilibrium's support for 3 simulations: $ N = 100 $, $ 400 $ and $ 900 $. Particles on the boundary have been relabelled to have consecutive indices starting from $ 1 $. The distances are shown in circles (connected by dotted, dash-dotted and dashed lines, respectively) and the corresponding thick lines represent their mean values. There are $ 31 $, $ 67 $ and $ 98 $ boundary points for the 3 simulations, with mean distances to centre of $ 0.5109 $, $ 0.5330 $ and $ 0.5415 $, and relative standard deviations of $ 0.55\% $, $ 0.47\% $ and $ 0.20\% $, respectively. The results strongly suggest that the continuum equilibrium is supported on a geodesic disk of radius $ R $ given by (71) (this value has been indicated as a thick solid line; $ R \approx 0.5570 $)

Figure 8.  Numerical explorations of equilibria of model (1) on $ H $ with the power-law interaction potential (75) (plots (a)-(c))) and the Morse-type potential (76)-(77) (plots (d)-(f)). (a) $ p = 0.5 $, $ q = 5 $. Equilibrium density supported on an annular region. (b) $ p = 1 $, $ q = 8 $. Concentration on a geodesic circle (ring). (c) $ p = 6 $, $ q = 7.5 $. Equilibrium consists of a delta accumulation on three points. (d) $ C = 1.2 $, $ l = 0.75 $, $ s = 2 $. Equilibrium supported on a geodesic circle with a delta accumulation at the centre. (e) $ C = 1.2 $, $ l = 0.75 $, $ s = 1.8 $. Concentration on a ring with a continuous density supported on a concentric interior disk. (f) $ C = 0.6 $, $ l = 0.5 $, $ s = 1.5 $. Concentration on a ring with a continuous density inside

Figure 9.  Numerical investigations with power-law potentials for the aggregation model on $ SO(3) $. The rotation matrices are plotted in angle-axis representation: the angles are shown on the left and the unit vectors are shown on the right plot, respectively. The initial states have been randomly generated (see text for details). The final states are marked by stars. (a)-(b) $ p = 5 $, $ q = 10 $. Aggregation in four points on $ SO(3) $ (in plot (a) we reordered the angles of the final state for a better visualization). (c)-(d) $ q = 2 $ and no repulsive term. Aggregation in one point on $ SO(3) $. The solution corresponds to a consensus in which all agents have synchronized their states