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Aggregation and disaggregation of active particles on the unit sphere with time-dependent frequencies

  • * Corresponding author: Jeongho Kim

    * Corresponding author: Jeongho Kim

The work of J. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science and ICT (NRF-2020R1A4A3079066)

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  • We introduce an active swarming model on the sphere which contains additional temporal dynamics for the natural frequency, inspired from the recently introduced modified Kuramoto model, where the natural frequency has its own dynamics. For the attractive interacting particle system, we provide a sufficient framework that leads to the asymptotic aggregation, i.e., all the particles are aggregated to the single point and the natural frequencies also tend to a common value. On the other hand, for the repulsive interacting particle system, we present a sufficient condition for the disaggregation, i.e., the order parameter of the system decays to 0, which implies that the particles are uniformly distributed over the sphere asymptotically. Finally, we also provide several numerical simulation results that support the theoretical results of the paper.

    Mathematics Subject Classification: Primary: 34C15, 34D06; Secondary: 34C40.


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  • Figure 1.  Set of initial data confined in a quadrant of the unit sphere

    Figure 2.  Particle trajectories for the attractive case ($ \kappa = 1 $): Simulation for Case 1 (Left), Case 2 (Middle) and Case 3 (Right). The blue marks on the sphere denote the initial positions of particles, while the red marks illustrate the positions of particles at the terminal time $ t = 50 $. The trajectory for Case 3 with $ \gamma = 1 $ blows up in finite time

    Figure 3.  The dynamics of the diameter of $ \Omega $ and the order parameters for the attractive case ($ \kappa = 1 $) with Case 1 (Left), Case 2 (Middle) and Case 3 (Right). The graphs for Case 3 with $ \gamma = 1 $ again blows up

    Figure 4.  Trajectory of the $ x_1(t) $ for the attractive case ($ \kappa = 1 $). Trajectory of the Case 1 (Left), trajectory for Case 2 (Middle) and trajectory for Case 3 (Right)

    Figure 5.  Numerical simulations for the repulsive case ($ \kappa = -1 $): Simulation for Case 1 (Left), Case 2 (Middle) and Case 3 (Right). The blue marks on sphere denote the initial positions of particles, while the red marks illustrate the positions of particles at the terminal time $ t = 50 $

    Figure 6.  The dynamics of the diameter of $ \Omega $ and the order parameters for the repulsive case ($ \kappa = -1 $) with Case 1 (Left), Case 2 (Middle) and Case 3 (Right). The graphs for Case 3 with $ \gamma = 1 $ again blows up

    Figure 7.  Trajectory of the $ x_1(t) $ for the repulsive case ($ \kappa = -1 $). Trajectory of the Case 1 (Left), trajectory for Case 2 (Middle) and trajectory for case 3 (Right)

    Table 1.  Choices for $ (C,\gamma_0) $ for each case

    $ C $ $ \gamma_0 $
    $ (\mathcal{C}_1) $ $ D({\bf\Omega}^0) $ $ \mu\Gamma_ {\rm{Lip}} $
    $ (\mathcal{C}_2) $ $ D({\bf\Omega}^0)\exp\left( \frac{2\mu \|\Omega_c^0\|_ { {\rm{F}}}}{\gamma} \right) $ 0
    $ (\mathcal{C}_3) $ $ D({\bf\Omega}^0) $ $ \max\left\{\frac{\mu \|\Psi\|_ { {\rm{F}}} (f(O,O)+\omega_2( D( {\bf\Omega}^0 )) )}{\min_{1\le i\le N}\;\;\|\Omega_i^0\|_ { {\rm{F}}} },2\mu\omega_1(R)\|\Psi\|_ { {\rm{F}}} \right\} $
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