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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 and Related Models, 2019, 12 (2) : 411-444. doi: 10.3934/krm.2019018
References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. 

[2]

T. Aoki and T. Aoyagi, Coevolution of phases and connection strengths in a network of phase oscillators, Phys. Rev. Lett., 102 (2009), 034101.

[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. 

[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.

[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.

[6]

J. ChoS.-Y. HaF. HuangC. 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.

[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.

[8]

Y.-P. ChoiS.-Y. HaS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

S.-Y. HaZ. 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.

[16]

S.-Y. HaD. KoJ. 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.

[17]

S.-Y. HaH. 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.

[18]

S.-Y. HaJ. LeeZ. 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. 

[19]

S.-Y. HaS. 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.

[20]

D. O. Hebb, The Organization of Behavior, Wiley, New York, 1949.

[21]

A. JadbabaieN. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2004), 4296-4301. 

[22]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin. 1984. doi: 10.1007/978-3-642-69689-3.

[23]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.

[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.

[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.

[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.

[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.

[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. 

[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.

[30]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.

[31]

Q. Ren and J. Zhao, Adaptive coupling and enhanced synchronization in coupled phase oscillators, Phys. Review E, 76 (2007), 016207.

[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.

[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.

[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.

[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.

[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. 

[37]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

show all references

References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. 

[2]

T. Aoki and T. Aoyagi, Coevolution of phases and connection strengths in a network of phase oscillators, Phys. Rev. Lett., 102 (2009), 034101.

[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. 

[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.

[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.

[6]

J. ChoS.-Y. HaF. HuangC. 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.

[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.

[8]

Y.-P. ChoiS.-Y. HaS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

S.-Y. HaZ. 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.

[16]

S.-Y. HaD. KoJ. 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.

[17]

S.-Y. HaH. 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.

[18]

S.-Y. HaJ. LeeZ. 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. 

[19]

S.-Y. HaS. 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.

[20]

D. O. Hebb, The Organization of Behavior, Wiley, New York, 1949.

[21]

A. JadbabaieN. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2004), 4296-4301. 

[22]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin. 1984. doi: 10.1007/978-3-642-69689-3.

[23]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.

[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.

[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.

[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.

[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.

[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. 

[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.

[30]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.

[31]

Q. Ren and J. Zhao, Adaptive coupling and enhanced synchronization in coupled phase oscillators, Phys. Review E, 76 (2007), 016207.

[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.

[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.

[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.

[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.

[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. 

[37]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

Figure 1.  Vector field of (60) for $\mu = \gamma = 1$
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