# American Institute of Mathematical Sciences

July & August  2021, 20(7&8): 2613-2641. doi: 10.3934/cpaa.2021046

## Collective behaviors of the Lohe Hermitian sphere model with inertia

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 2 Korea Institute for Advanced Study, Hoegiro 85, 02455, Seoul, Republic of Korea 3 Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

Dedicated to the celebration of the 80th birthday of Professor Shuxing Chen

Received  September 2020 Revised  February 2021 Published  July & August 2021 Early access  March 2021

Fund Project: The work of S.-Y.Ha is supported by NRF-2020R1A2C3A01003881, the work of M. Kang was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP)(2016K2A9A2A13003815), and the work of H. Park was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)

We present a second-order extension of the first-order Lohe Hermitian sphere (LHS) model and study its emergent asymptotic dynamics. Our proposed model incorporates an inertial effect as a second-order extension. The inertia term can generate an oscillatory behavior of particle trajectory in a small time interval(initial layer) which causes a technical difficulty for the application of monotonicity-based arguments. For emergent estimates, we employ two-point correlation function which is defined as an inner product between positions of particles. For a homogeneous ensemble with the same frequency matrix, we provide two sufficient frameworks in terms of system parameters and initial data to show that two-point correlation functions tend to the unity which is exactly the same as the complete aggregation. In contrast, for a heterogeneous ensemble with distinct frequency matrices, we provide a sufficient framework in terms of system parameters and initial data, which makes two-point correlation functions be close to unity by increasing the principal coupling strength.

Citation: Seung-Yeal Ha, Myeongju Kang, Hansol Park. Collective behaviors of the Lohe Hermitian sphere model with inertia. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2613-2641. doi: 10.3934/cpaa.2021046
##### 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. [2] I. Barb$\check{a}$lat, Syst$\grave{e}$mes d$\acute{e}$quations diff$\acute{e}$rentielles d$\bar{o}$oscillations non Lin$\acute{e}$aires, Rev. Math. Pures Appl., 4 (1959), 267-270. [3] J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562. [4] 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. doi: 10.1063/1.4878117. [5] Y. P. Choi, S. Y. Ha and S. B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D: Nonlinear Phenomena, 1 (2011), 32-44.  doi: 10.1016/j.physd.2010.08.004. [6] 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. [7] S. Y. Ha and D. Kim, A second-order particle swarm model on a sphere and emergent dynamics, SIAM J. Appl. Dyn. Syst., 18 (2019), 80-116.  doi: 10.1137/18M1205996. [8] S. Y. Ha, D. Kim, J. Lee and S. E. Noh, Particle and kinetic models for swarming particles on a sphere and their stability properties, J. Stat. Phys., 174 (2019), 622-655.  doi: 10.1007/s10955-018-2169-8. [9] S. Y. Ha, M. Kang and H. Park, Emergent dynamics of the Lohe Hermitian sphere model with frustration, submitted. [10] S. Y. Ha and H. Park, Complete aggregation of the Lohe tensor model with the same free flow, J. Math. Phys., 61 (2020), 102702 doi: 10.1063/5.0007292. [11] S. Y. Ha and H. Park, From the Lohe tensor model to the Lohe Hermitian sphere model and emergent dynamics, SIAM J. Appl. Dyn. Syst., 19 (2020), 1312-1342.  doi: 10.1137/19M1288553. [12] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. [13] Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, In International Symposium on Mathematical Problems in Mathematical Physics. Lecture Notes in Theoretical Physics 30 (1975), 420–422. [14] M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301. doi: 10.1088/1751-8113/43/46/465301. [15] M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101. doi: 10.1088/1751-8113/42/39/395101. [16] J. Markdahl, J. Thunberg and J. Gonąlves, Towards almost global synchronization on the Stiefel manifold, To appear in the proceedings of the 57th IEEE Conference on Decision and Control. Miami, FL, USA, 2017. doi: 10.1109/tac.2017.2752799. [17] J. Markdahl, J. Thunberg and J. Gonąlves, High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally, Automatica, 113 (2020), 108736. doi: 10.1016/j.automatica.2019.108736. [18] 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. [19] C. S. Peskin, Mathematical aspect of heart physiology, Courant Institute of Mathematical Sciences, New York, 1975. [20] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743. [21] L. M. Ritchie, M. A. Lohe and A. G. Williams, Synchronization of relativistic particles in the hyperbolic Kuramoto model, Chaos, 28 (2018), 053116 doi: 10.1063/1.5021701. [22] 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. [23] R. Sknepnek and S. Henkes, Active swarms on a sphere, Phys. Rev. E, 2 (2015), 022306. [24] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. [25] J. Zhang, J. Zhu and C. Qian, On equilibria and consensus of the Lohe model with identical oscillators, SIAM J. Appl. Dyn. Syst., 17 (2018), 1716-1741.  doi: 10.1137/17M112765X. [26] J. Zhu, Synchronization of Kuramoto model in a high-dimensional linear space, Phys. Lett. A, 377 (2013), 2939-2943.  doi: 10.1016/j.physleta.2013.09.010.

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##### 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. [2] I. Barb$\check{a}$lat, Syst$\grave{e}$mes d$\acute{e}$quations diff$\acute{e}$rentielles d$\bar{o}$oscillations non Lin$\acute{e}$aires, Rev. Math. Pures Appl., 4 (1959), 267-270. [3] J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562. [4] 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. doi: 10.1063/1.4878117. [5] Y. P. Choi, S. Y. Ha and S. B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D: Nonlinear Phenomena, 1 (2011), 32-44.  doi: 10.1016/j.physd.2010.08.004. [6] 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. [7] S. Y. Ha and D. Kim, A second-order particle swarm model on a sphere and emergent dynamics, SIAM J. Appl. Dyn. Syst., 18 (2019), 80-116.  doi: 10.1137/18M1205996. [8] S. Y. Ha, D. Kim, J. Lee and S. E. Noh, Particle and kinetic models for swarming particles on a sphere and their stability properties, J. Stat. Phys., 174 (2019), 622-655.  doi: 10.1007/s10955-018-2169-8. [9] S. Y. Ha, M. Kang and H. Park, Emergent dynamics of the Lohe Hermitian sphere model with frustration, submitted. [10] S. Y. Ha and H. Park, Complete aggregation of the Lohe tensor model with the same free flow, J. Math. Phys., 61 (2020), 102702 doi: 10.1063/5.0007292. [11] S. Y. Ha and H. Park, From the Lohe tensor model to the Lohe Hermitian sphere model and emergent dynamics, SIAM J. Appl. Dyn. Syst., 19 (2020), 1312-1342.  doi: 10.1137/19M1288553. [12] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. [13] Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, In International Symposium on Mathematical Problems in Mathematical Physics. Lecture Notes in Theoretical Physics 30 (1975), 420–422. [14] M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301. doi: 10.1088/1751-8113/43/46/465301. [15] M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101. doi: 10.1088/1751-8113/42/39/395101. [16] J. Markdahl, J. Thunberg and J. Gonąlves, Towards almost global synchronization on the Stiefel manifold, To appear in the proceedings of the 57th IEEE Conference on Decision and Control. Miami, FL, USA, 2017. doi: 10.1109/tac.2017.2752799. [17] J. Markdahl, J. Thunberg and J. Gonąlves, High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally, Automatica, 113 (2020), 108736. doi: 10.1016/j.automatica.2019.108736. [18] 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. [19] C. S. Peskin, Mathematical aspect of heart physiology, Courant Institute of Mathematical Sciences, New York, 1975. [20] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743. [21] L. M. Ritchie, M. A. Lohe and A. G. Williams, Synchronization of relativistic particles in the hyperbolic Kuramoto model, Chaos, 28 (2018), 053116 doi: 10.1063/1.5021701. [22] 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. [23] R. Sknepnek and S. Henkes, Active swarms on a sphere, Phys. Rev. E, 2 (2015), 022306. [24] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. [25] J. Zhang, J. Zhu and C. Qian, On equilibria and consensus of the Lohe model with identical oscillators, SIAM J. Appl. Dyn. Syst., 17 (2018), 1716-1741.  doi: 10.1137/17M112765X. [26] J. Zhu, Synchronization of Kuramoto model in a high-dimensional linear space, Phys. Lett. A, 377 (2013), 2939-2943.  doi: 10.1016/j.physleta.2013.09.010.
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