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April  2019, 12(2): 411-444. doi: 10.3934/krm.2019018

## Emergence of aggregation in the swarm sphere model with adaptive coupling laws

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea 2 Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea 3 Seoul National University, Dept. of Math. Sciences, Seoul 08826, Republic of Korea 4 Myongji University, Dept. of Mathematics, Yong-In 17058, Republic of Korea

* Corresponding author: Se Eun Noh

Received  June 2018 Published  November 2018

We present aggregation estimates for the swarm sphere model equipped with the adaptive coupling laws on a sphere. The temporal evolution of coupling strength is determined by a feedback rule incorporating the balance between relative spatial variations and linear damping. For the analytical treatment, we employ two adaptive feedback laws, namely anti-Hebbian and Hebbian laws. For the anti-Hebbian law, we provide a sufficient framework leading to the complete aggregation in which all particles aggregate to the same position and behave like one big point cluster asymptotically. Our frameworks are given in terms of the initial positions and the coupling strengths. For the Hebbian law, we provide proper subsets of the basin of attractions for the complete aggregation and bi-polar aggregation where particles aggregate to the north pole and south pole simultaneously. We also present a uniform $\ell_p$-stability of the swarm sphere model with an adaptive coupling with respect to the initial data when the complete aggregation occurs exponentially fast.

Citation: Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergence of aggregation in the swarm sphere model with adaptive coupling laws. Kinetic & Related Models, 2019, 12 (2) : 411-444. doi: 10.3934/krm.2019018
##### References:
 [1] J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. Google Scholar [2] T. Aoki and T. Aoyagi, Coevolution of phases and connection strengths in a network of phase oscillators, Phys. Rev. Lett., 102 (2009), 034101.Google Scholar [3] I. Barb$\check{a}$lat, Syst$\grave{e}$mes d$\acute{e}$quations diff$\acute{e}$rentielles d oscillations non Lin$\acute{e}$aires, Rev. Math. Pures Appl., 4 (1959), 267-270. Google Scholar [4] Bronski, J. C., DeVille, L., and M. J. Park, Fully Synchronous Solutions and the Synchronization Phase Transition for the Finite N Kuramoto Model, Chaos, 22 (2012), 033133, 17pp. doi: 10.1063/1.4745197. Google Scholar [5] D. Chi, S.-H. Choi and S.-Y. Ha, Emergent behaviors of a holonomic particle system on a sphere, J. Math. Phys., 55 (2014), 052703, 18pp. doi: 10.1063/1.4878117. Google Scholar [6] J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Method. in Applied Sci., 26 (2016), 1191-1218. doi: 10.1142/S0218202516500287. Google Scholar [7] S.-H. Choi and S.-Y. Ha, Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM J. Appl. Dyn. Syst., 13 (2014), 1417-1441. doi: 10.1137/140961699. Google Scholar [8] Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754. doi: 10.1016/j.physd.2011.11.011. Google Scholar [9] N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automatic Control, 54 (2009), 353-357. doi: 10.1109/TAC.2008.2007884. Google Scholar [10] D. Cumin and C. P. Unsworth, Generalizing the Kuramoto model for the study of neuronal synchronization in the brain, Phys. D, 226 (2007), 181-196. doi: 10.1016/j.physd.2006.12.004. Google Scholar [11] J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480. doi: 10.4310/CMS.2013.v11.n2.a7. Google Scholar [12] F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564. doi: 10.1016/j.automatica.2014.04.012. Google Scholar [13] F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099. doi: 10.1137/10081530X. Google Scholar [14] A. Gushchin, E. Mallada and A. Tang, Synchronization of Phase-Coupled Oscillators with Plastic Coupling Strength, Proceedings of Information Theory and Applications Workshop. 2015.Google Scholar [15] S.-Y. Ha, Z. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070. doi: 10.1016/j.jde.2013.07.013. Google Scholar [16] S.-Y. Ha, D. Ko, J. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267. doi: 10.4171/EMSS/17. Google Scholar [17] S.-Y. Ha, H. Kim and S. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091. doi: 10.4310/CMS.2016.v14.n4.a10. Google Scholar [18] S.-Y. Ha, J. Lee, Z. Li and J. Park, Emergent dynamics of Kuramoto oscillators with adaptive couplings: conservation law and fast learning, SIAM J. Appl. Dyn. Syst., 17 (2018), 1560-1588. Google Scholar [19] S.-Y. Ha, S. E. Noh and J. Park, Syncrhonization of kuramoto oscillators with adaptive couplings, SIAM J. Appl. Dyn. Syst., 15 (2016), 162-194. doi: 10.1137/15M101484X. Google Scholar [20] D. O. Hebb, The Organization of Behavior, Wiley, New York, 1949.Google Scholar [21] A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2004), 4296-4301. Google Scholar [22] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin. 1984. doi: 10.1007/978-3-642-69689-3. Google Scholar [23] Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.Google Scholar [24] M. A. Lohe, Quantum synchronization over quantum networks J. Phys. A: Math. Theor. 43 (2010), 465301, 20pp. doi: 10.1088/1751-8113/43/46/465301. Google Scholar [25] M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101, 25 pp. doi: 10.1088/1751-8113/42/39/395101. Google Scholar [26] R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), 1645-1662. doi: 10.1137/0150098. Google Scholar [27] R. K. Niyogi and L. Q. English, Learning-rate-dependent clustering and self-development in a network of coupled phase oscillators. Phys. Rev. E, 80 (2009), 066213.Google Scholar [28] R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, Proc. of the 45th IEEE conference on Decision and Control, (2006), 5060-5066. Google Scholar [29] C. B. Piccallo and H. Riecke, Adaptive oscillator networks with conserved overall coupling: Sequential firing and near-synchronized states, Phys. Review E, 83 (2011), 036206, 12pp. doi: 10.1103/PhysRevE.83.036206. Google Scholar [30] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743. Google Scholar [31] Q. Ren and J. Zhao, Adaptive coupling and enhanced synchronization in coupled phase oscillators, Phys. Review E, 76 (2007), 016207.Google Scholar [32] L. Scardovi, Clustering and synchronization in phase models with state dependent coupling, 49th IEEE Conference on Decision and Control, December 15-17, 2010, Hilton Atlanta Hotel, Atlanta, GA, USA.Google Scholar [33] P. Seliger, S. C. Young and L. S. Tsimring, Plasticity and learning in a network of coupled phase oscillators, Phys. Review E, 65 (2002), 041906, 7pp. doi: 10.1103/PhysRevE.65.041906. Google Scholar [34] S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4. Google Scholar [35] D. Taylor, E. Ott and J. G. Restrepo, Spontaneous synchronization of coupled oscillator systems with frequency adaptation, Phys. Rev. E, 81 (2010), 046214, 8pp. doi: 10.1103/PhysRevE.81.046214. Google Scholar [36] J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166. Google Scholar [37] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. Google Scholar

show all references

##### References:
 [1] J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. Google Scholar [2] T. Aoki and T. Aoyagi, Coevolution of phases and connection strengths in a network of phase oscillators, Phys. Rev. Lett., 102 (2009), 034101.Google Scholar [3] I. Barb$\check{a}$lat, Syst$\grave{e}$mes d$\acute{e}$quations diff$\acute{e}$rentielles d oscillations non Lin$\acute{e}$aires, Rev. Math. Pures Appl., 4 (1959), 267-270. Google Scholar [4] Bronski, J. C., DeVille, L., and M. J. Park, Fully Synchronous Solutions and the Synchronization Phase Transition for the Finite N Kuramoto Model, Chaos, 22 (2012), 033133, 17pp. doi: 10.1063/1.4745197. Google Scholar [5] D. Chi, S.-H. Choi and S.-Y. Ha, Emergent behaviors of a holonomic particle system on a sphere, J. Math. Phys., 55 (2014), 052703, 18pp. doi: 10.1063/1.4878117. Google Scholar [6] J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Method. in Applied Sci., 26 (2016), 1191-1218. doi: 10.1142/S0218202516500287. Google Scholar [7] S.-H. Choi and S.-Y. Ha, Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM J. Appl. Dyn. Syst., 13 (2014), 1417-1441. doi: 10.1137/140961699. Google Scholar [8] Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754. doi: 10.1016/j.physd.2011.11.011. Google Scholar [9] N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automatic Control, 54 (2009), 353-357. doi: 10.1109/TAC.2008.2007884. Google Scholar [10] D. Cumin and C. P. Unsworth, Generalizing the Kuramoto model for the study of neuronal synchronization in the brain, Phys. D, 226 (2007), 181-196. doi: 10.1016/j.physd.2006.12.004. Google Scholar [11] J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480. doi: 10.4310/CMS.2013.v11.n2.a7. Google Scholar [12] F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564. doi: 10.1016/j.automatica.2014.04.012. Google Scholar [13] F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099. doi: 10.1137/10081530X. Google Scholar [14] A. Gushchin, E. Mallada and A. Tang, Synchronization of Phase-Coupled Oscillators with Plastic Coupling Strength, Proceedings of Information Theory and Applications Workshop. 2015.Google Scholar [15] S.-Y. Ha, Z. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070. doi: 10.1016/j.jde.2013.07.013. Google Scholar [16] S.-Y. Ha, D. Ko, J. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267. doi: 10.4171/EMSS/17. Google Scholar [17] S.-Y. Ha, H. Kim and S. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091. doi: 10.4310/CMS.2016.v14.n4.a10. Google Scholar [18] S.-Y. Ha, J. Lee, Z. Li and J. Park, Emergent dynamics of Kuramoto oscillators with adaptive couplings: conservation law and fast learning, SIAM J. Appl. Dyn. Syst., 17 (2018), 1560-1588. Google Scholar [19] S.-Y. Ha, S. E. Noh and J. Park, Syncrhonization of kuramoto oscillators with adaptive couplings, SIAM J. Appl. Dyn. Syst., 15 (2016), 162-194. doi: 10.1137/15M101484X. Google Scholar [20] D. O. Hebb, The Organization of Behavior, Wiley, New York, 1949.Google Scholar [21] A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2004), 4296-4301. Google Scholar [22] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin. 1984. doi: 10.1007/978-3-642-69689-3. Google Scholar [23] Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.Google Scholar [24] M. A. Lohe, Quantum synchronization over quantum networks J. Phys. A: Math. Theor. 43 (2010), 465301, 20pp. doi: 10.1088/1751-8113/43/46/465301. Google Scholar [25] M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101, 25 pp. doi: 10.1088/1751-8113/42/39/395101. Google Scholar [26] R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), 1645-1662. doi: 10.1137/0150098. Google Scholar [27] R. K. Niyogi and L. Q. English, Learning-rate-dependent clustering and self-development in a network of coupled phase oscillators. Phys. Rev. E, 80 (2009), 066213.Google Scholar [28] R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, Proc. of the 45th IEEE conference on Decision and Control, (2006), 5060-5066. Google Scholar [29] C. B. Piccallo and H. Riecke, Adaptive oscillator networks with conserved overall coupling: Sequential firing and near-synchronized states, Phys. Review E, 83 (2011), 036206, 12pp. doi: 10.1103/PhysRevE.83.036206. Google Scholar [30] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743. Google Scholar [31] Q. Ren and J. Zhao, Adaptive coupling and enhanced synchronization in coupled phase oscillators, Phys. Review E, 76 (2007), 016207.Google Scholar [32] L. Scardovi, Clustering and synchronization in phase models with state dependent coupling, 49th IEEE Conference on Decision and Control, December 15-17, 2010, Hilton Atlanta Hotel, Atlanta, GA, USA.Google Scholar [33] P. Seliger, S. C. Young and L. S. Tsimring, Plasticity and learning in a network of coupled phase oscillators, Phys. Review E, 65 (2002), 041906, 7pp. doi: 10.1103/PhysRevE.65.041906. Google Scholar [34] S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4. Google Scholar [35] D. Taylor, E. Ott and J. G. Restrepo, Spontaneous synchronization of coupled oscillator systems with frequency adaptation, Phys. Rev. E, 81 (2010), 046214, 8pp. doi: 10.1103/PhysRevE.81.046214. Google Scholar [36] J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166. Google Scholar [37] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. Google Scholar
Vector field of (60) for $\mu = \gamma = 1$
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