American Institute of Mathematical Sciences

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September  2019, 11(3): 397-426. doi: 10.3934/jgm.2019020

Self-organization on Riemannian manifolds

Razvan C. Fetecau , and  Beril Zhang

Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada

* Corresponding author: Razvan C. Fetecau

Received  August 2018 Revised  July 2019 Published  August 2019

Fund Project: The first author is supported by NSERC grant RGPIN-341834.

We consider an aggregation model that consists of an active transport equation for the macroscopic population density, where the velocity has a nonlocal functional dependence on the density, modelled via an interaction potential. We set up the model on general Riemannian manifolds and provide a framework for constructing interaction potentials which lead to equilibria that are constant on their supports. We consider such potentials for two specific cases (the two-dimensional sphere and the two-dimensional hyperbolic space) and investigate analytically and numerically the long-time behaviour and equilibrium solutions of the aggregation model on these manifolds. Equilibria obtained numerically with other interaction potentials and an application of the model to aggregation on the rotation group $ SO(3) $ are also presented.

Keywords: Swarming on manifolds, uniform densities, global attractors, hyperbolic space.
Mathematics Subject Classification: Primary: 35R01, 34C40; Secondary: 35B36, 35Q70.
Citation: Razvan C. Fetecau, Beril Zhang. Self-organization on Riemannian manifolds. Journal of Geometric Mechanics, 2019, 11 (3) : 397-426. doi: 10.3934/jgm.2019020
References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

B. Afsari, Riemannian ${L}^{p}$ center of mass: Existence, uniqueness, and convexity, Proc. Amer. Math. Soc., 139 (2011), 655-673.  doi: 10.1090/S0002-9939-2010-10541-5.

[3]

B. Afsari, R. Tron and R. Vidal, On the convergence of gradient descent for finding the Riemannian center of mass, SIAM J. Control Optim., 51 (2013), 2230-2260.  doi: 10.1137/12086282X.

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.

[5]

D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088.  doi: 10.1007/s00205-013-0644-6.

[6]

D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul, Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), 5-25.  doi: 10.1016/j.physd.2012.10.002.

[7]

A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250.  doi: 10.1137/100804504.

[8]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.  doi: 10.1088/0951-7715/22/3/009.

[9]

A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in ${\bf{R}}^n$, Comm. Math. Phys., 274 (2007), 717-735.  doi: 10.1007/s00220-007-0288-1.

[10]

A. L. Bertozzi, T. Laurent and L. Flavien, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 1140005, 39pp.  doi: 10.1142/S0218202511400057.

[11]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334.

[12]

M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380.  doi: 10.1016/j.jde.2005.07.025.

[13]

M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.  doi: 10.3934/nhm.2008.3.749.

[14]

S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Selforganization in Biological Systems, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003, Reprint of the 2001 original.

[15]

J. A. Cañizo, J. A. Carrillo and F. S. Patacchini, Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), 1197-1217.  doi: 10.1007/s00205-015-0852-3.

[16]

J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Hyperbolic geometry, in Flavors of Geometry, vol. 31 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1997, 59-115.

[17]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.  doi: 10.1215/00127094-2010-211.

[18]

J. A. Carrillo, Y. Huang and S. Martin, Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578.  doi: 10.1017/S0956792514000126.

[19]

J. A. Carrillo, D. Slepčev and L. Wu, Nonlocal-interaction equations on uniformly prox-regular sets, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1209-1247.  doi: 10.3934/dcds.2016.36.1209.

[20]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[21]

J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, vol. 553 of CISM Courses and Lect., Springer, Vienna, 2014, 1–46. doi: 10.1007/978-3-7091-1785-9_1.

[22]

R. Choksi, R. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1283-1305.  doi: 10.1016/j.anihpc.2014.09.004.

[23]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.  doi: 10.1016/j.physd.2007.05.007.

[24]

J. Cortés and M. Egerstedt, Coordinated control of multi-robot systems: A survey, SICE Journal of Control, Measurement, and System Integration, 10 (2017), 495-503. 

[25]

I. D. Couzin, J. Krause, R. James, G. Ruxton and N. R. Franks, Collective memory and spatial sorting in animal groups, J. Theor. Biol., 218 (2002), 1-11.  doi: 10.1006/jtbi.2002.3065.

[26]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[27]

A. Czirók, A.-L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Physical Review Letters, 82 (1999), 209-212. 

[28]

R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci., 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104.

[29]

J. H. M. Evers, R. C. Fetecau and L. Ryzhik, Anisotropic interactions in a first-order aggregation model, Nonlinearity, 28 (2015), 2847-2871.  doi: 10.1088/0951-7715/28/8/2847.

[30]

K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291.  doi: 10.1142/S0218202510004921.

[31]

R. C. Fetecau and Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64.  doi: 10.1016/j.physd.2012.11.004.

[32]

R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[33]

R. C. Fetecau and M. Kovacic, Swarm equilibria in domains with boundaries, SIAM Journal on Applied Dynamical Systems, 16 (2017), 1260-1308.  doi: 10.1137/17M1123900.

[34]

V. Gazi and K. M. Passino, Stability analysis of swarms, IEEE Trans. Automat. Control, 48 (2003), 692-697.  doi: 10.1109/TAC.2003.809765.

[35]

J. Haile, Molecular Dynamics Simulation: Elementary Methods, John Wiley and Sons, Inc., New York, 1992.

[36]

D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Phys Rev Lett., 95 (2005), 226106.  doi: 10.1103/PhysRevLett.95.226106.

[37]

J. L. A. Jadbabaie and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.

[38]

M. Ji and M. Egerstedt, Distributed coordination control of multi-agent systems while preserving connectedness, IEEE Trans. Robot., 23 (2007), 693-703. 

[39]

H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977), 509-541.  doi: 10.1002/cpa.3160300502.

[40]

Y. Kimura, Vortex motion on surfaces with constant curvature, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 245-259.  doi: 10.1098/rspa.1999.0311.

[41]

T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Communications, 84 (2011), 015203(R).

[42]

W. Kühnel, Differential Geometry, vol. 16 of Student Mathematical Library, American Mathematical Society, Providence, RI, 2002.

[43]

A. J. Leverentz, C. M. Topaz and A. J. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908.  doi: 10.1137/090749037.

[44]

W. Li, Collective motion of swarming agents evolving on a sphere manifold: A fundamental framework and characterization, Scientific Reports, 5 (2015), 13603.  doi: 10.1038/srep13603.

[45]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.

[46]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[47]

P. Petersen, Riemannian Geometry, vol. 171 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2006.

[48]

G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Math. Acad. Sci. Paris, 345 (2007), 151-154.  doi: 10.1016/j.crma.2007.06.018.

[49]

R. Sepulchre, Consensus on nonlinear spaces, Annual Reviews in Control, 35 (2011), 56-64. 

[50]

R. Simione, D. Slepčev and I. Topaloglu, Existence of ground states of nonlocal-interaction energies, J. Stat. Phys., 159 (2015), 972-986.  doi: 10.1007/s10955-015-1215-z.

[51]

R. Tron, B. Afsari and R. Vidal, Intrinsic consensus on $SO(3)$ with almost-global convergence, Proceedings of the 51st IEEE Conference on Decision and Control, 2013, 2052–2058. doi: 10.1109/CDC.2012.6426677.

[52]

J. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics, J. Nonlinear Sci., 22 (2012), 935-959.  doi: 10.1007/s00332-012-9132-7.

[53]

J. von Brecht, D. Uminsky, T. Kolokolnikov and A. Bertozzi, Predicting pattern formation in particle interactions, Math. Models Methods Appl. Sci., 22 (2012), 1140002, 31pp.  doi: 10.1142/S0218202511400021.

[54]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries, Comm. Partial Differential Equations, 40 (2015), 1241-1281.  doi: 10.1080/03605302.2015.1015033.

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

B. Afsari, Riemannian ${L}^{p}$ center of mass: Existence, uniqueness, and convexity, Proc. Amer. Math. Soc., 139 (2011), 655-673.  doi: 10.1090/S0002-9939-2010-10541-5.

[3]

B. Afsari, R. Tron and R. Vidal, On the convergence of gradient descent for finding the Riemannian center of mass, SIAM J. Control Optim., 51 (2013), 2230-2260.  doi: 10.1137/12086282X.

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.

[5]

D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088.  doi: 10.1007/s00205-013-0644-6.

[6]

D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul, Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), 5-25.  doi: 10.1016/j.physd.2012.10.002.

[7]

A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250.  doi: 10.1137/100804504.

[8]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.  doi: 10.1088/0951-7715/22/3/009.

[9]

A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in ${\bf{R}}^n$, Comm. Math. Phys., 274 (2007), 717-735.  doi: 10.1007/s00220-007-0288-1.

[10]

A. L. Bertozzi, T. Laurent and L. Flavien, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 1140005, 39pp.  doi: 10.1142/S0218202511400057.

[11]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334.

[12]

M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380.  doi: 10.1016/j.jde.2005.07.025.

[13]

M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.  doi: 10.3934/nhm.2008.3.749.

[14]

S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Selforganization in Biological Systems, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003, Reprint of the 2001 original.

[15]

J. A. Cañizo, J. A. Carrillo and F. S. Patacchini, Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), 1197-1217.  doi: 10.1007/s00205-015-0852-3.

[16]

J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Hyperbolic geometry, in Flavors of Geometry, vol. 31 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1997, 59-115.

[17]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.  doi: 10.1215/00127094-2010-211.

[18]

J. A. Carrillo, Y. Huang and S. Martin, Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578.  doi: 10.1017/S0956792514000126.

[19]

J. A. Carrillo, D. Slepčev and L. Wu, Nonlocal-interaction equations on uniformly prox-regular sets, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1209-1247.  doi: 10.3934/dcds.2016.36.1209.

[20]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[21]

J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, vol. 553 of CISM Courses and Lect., Springer, Vienna, 2014, 1–46. doi: 10.1007/978-3-7091-1785-9_1.

[22]

R. Choksi, R. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1283-1305.  doi: 10.1016/j.anihpc.2014.09.004.

[23]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.  doi: 10.1016/j.physd.2007.05.007.

[24]

J. Cortés and M. Egerstedt, Coordinated control of multi-robot systems: A survey, SICE Journal of Control, Measurement, and System Integration, 10 (2017), 495-503. 

[25]

I. D. Couzin, J. Krause, R. James, G. Ruxton and N. R. Franks, Collective memory and spatial sorting in animal groups, J. Theor. Biol., 218 (2002), 1-11.  doi: 10.1006/jtbi.2002.3065.

[26]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[27]

A. Czirók, A.-L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Physical Review Letters, 82 (1999), 209-212. 

[28]

R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci., 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104.

[29]

J. H. M. Evers, R. C. Fetecau and L. Ryzhik, Anisotropic interactions in a first-order aggregation model, Nonlinearity, 28 (2015), 2847-2871.  doi: 10.1088/0951-7715/28/8/2847.

[30]

K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291.  doi: 10.1142/S0218202510004921.

[31]

R. C. Fetecau and Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64.  doi: 10.1016/j.physd.2012.11.004.

[32]

R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[33]

R. C. Fetecau and M. Kovacic, Swarm equilibria in domains with boundaries, SIAM Journal on Applied Dynamical Systems, 16 (2017), 1260-1308.  doi: 10.1137/17M1123900.

[34]

V. Gazi and K. M. Passino, Stability analysis of swarms, IEEE Trans. Automat. Control, 48 (2003), 692-697.  doi: 10.1109/TAC.2003.809765.

[35]

J. Haile, Molecular Dynamics Simulation: Elementary Methods, John Wiley and Sons, Inc., New York, 1992.

[36]

D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Phys Rev Lett., 95 (2005), 226106.  doi: 10.1103/PhysRevLett.95.226106.

[37]

J. L. A. Jadbabaie and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.

[38]

M. Ji and M. Egerstedt, Distributed coordination control of multi-agent systems while preserving connectedness, IEEE Trans. Robot., 23 (2007), 693-703. 

[39]

H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977), 509-541.  doi: 10.1002/cpa.3160300502.

[40]

Y. Kimura, Vortex motion on surfaces with constant curvature, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 245-259.  doi: 10.1098/rspa.1999.0311.

[41]

T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Communications, 84 (2011), 015203(R).

[42]

W. Kühnel, Differential Geometry, vol. 16 of Student Mathematical Library, American Mathematical Society, Providence, RI, 2002.

[43]

A. J. Leverentz, C. M. Topaz and A. J. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908.  doi: 10.1137/090749037.

[44]

W. Li, Collective motion of swarming agents evolving on a sphere manifold: A fundamental framework and characterization, Scientific Reports, 5 (2015), 13603.  doi: 10.1038/srep13603.

[45]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.

[46]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[47]

P. Petersen, Riemannian Geometry, vol. 171 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2006.

[48]

G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Math. Acad. Sci. Paris, 345 (2007), 151-154.  doi: 10.1016/j.crma.2007.06.018.

[49]

R. Sepulchre, Consensus on nonlinear spaces, Annual Reviews in Control, 35 (2011), 56-64. 

[50]

R. Simione, D. Slepčev and I. Topaloglu, Existence of ground states of nonlocal-interaction energies, J. Stat. Phys., 159 (2015), 972-986.  doi: 10.1007/s10955-015-1215-z.

[51]

R. Tron, B. Afsari and R. Vidal, Intrinsic consensus on $SO(3)$ with almost-global convergence, Proceedings of the 51st IEEE Conference on Decision and Control, 2013, 2052–2058. doi: 10.1109/CDC.2012.6426677.

[52]

J. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics, J. Nonlinear Sci., 22 (2012), 935-959.  doi: 10.1007/s00332-012-9132-7.

[53]

J. von Brecht, D. Uminsky, T. Kolokolnikov and A. Bertozzi, Predicting pattern formation in particle interactions, Math. Models Methods Appl. Sci., 22 (2012), 1140002, 31pp.  doi: 10.1142/S0218202511400021.

[54]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries, Comm. Partial Differential Equations, 40 (2015), 1241-1281.  doi: 10.1080/03605302.2015.1015033.

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)
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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)
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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)
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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)
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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
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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))
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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 $)
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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
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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
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