Abstract
We present a first-order aggregation model for a homogeneous Lohe matrix ensemble with higher order couplings via a gradient flow approach. For homogeneous free flow with the same Hamiltonian, it is well known that the Lohe matrix model with cubic couplings can be recast as a gradient system with a potential which is a squared Frobenius norm of of averaged state. In this paper, we further derive a generalized Lohe matrix model with higher-order couplings via gradient flow approach for a polynomial potential. For the proposed model, we also provide a sufficient framework in terms of coupling strengths and initial data leading to the emergent dynamics of a homogeneous ensemble.
Similar content being viewed by others
References
Acebron, J.A., Bonilla, L.L., Pérez Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)
Barbǎlat, I.: Systèmes déquations différentielles dŌoscillations non Linéaires. Rev. Math. Pures Appl. 4, 267–270 (1959)
Benedetto, D., Caglioti, E., Montemagno, U.: On the complete phase synchronization for the Kuramoto model in the mean-field limit. Commun. Math. Sci. 13, 1775–1786 (2015)
Buck, J., Buck, E.: Biology of synchronous flashing of fireflies. Nature 211, 562–564 (1966)
Chen, B., Engelbrecht, J.R., Mirollo, R.: Hyperbolic geometry of Kuramoto oscillator networks. J. Phys. A 50, 355101 (2017)
Chen, B., Engelbrecht, J.R., Mirollo, R.: Dynamics of the Kuramoto–Sakaguchi oscillator network with asymmetric order parameter. Chaos 29, 013126 (2019)
Choi, S.-H., Ha, S.-Y.: Complete entrainment of Lohe oscillators under attractive and repulsive couplings. SIAM J. App. Dyn. Syst. 13, 1417–1441 (2013)
Choi, Y., Ha, S.-Y., Jung, S., Kim, Y.: Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model. Phys. D 241, 735–754 (2012)
Chopra, N., Spong, M.W.: On exponential synchronization of Kuramoto oscillators. IEEE Trans. Autom. Control 54, 353–357 (2009)
Daido, H.: Order function and macroscopic mutual entrainment in uniformly coupled limit-cycle oscillators. Prog. Theor. Phys. 88, 1213–1218 (1992)
DeVille, L.: Synchronization and stability for quantum Kuramoto. J. Stat. Phys. 174, 160–187 (2019)
Dong, J.-G., Xue, X.: Synchronization analysis of Kuramoto oscillators. Commun. Math. Sci. 11, 465–480 (2013)
Dörfler, F., Bullo, F.: On the critical coupling for Kuramoto oscillators. SIAM J. Appl. Dyn. Syst. 10, 1070–1099 (2011)
Dörfler, F., Bullo, F.: Synchronization in complex networks of phase oscillators: a survey. Automatica 50, 1539–1564 (2014)
Ha, S.-Y., Park, H.: Complete aggregation of the Lohe tensor model with the same free flow. J. Math. Phys. 61, 102702 (2020)
Ha, S.-Y., Park, H.: Emergent behaviors of Lohe tensor flocks. J. Stat. Phys. 178, 1268–1292 (2020)
Ha, S.-Y., Ryoo, S.W.: On the emergence and orbital stability of phase-locked states for the Lohe model. J. Stat. Phys. 163, 411–439 (2016)
Ha, S.-Y., Li, Z., Xue, X.: Formation of phase-locked states in a population of locally interacting Kuramoto oscillators. J. Differ. Equ. 255, 3053–3070 (2013)
Ha, S.-Y., Kim, H.W., Ryoo, S.W.: Emergence of phase-locked states for the Kuramoto model in a large coupling regime. Commun. Math. Sci. 14, 1073–1091 (2016)
Ha, S.-Y., Ko, D., Ryoo, S.W.: Emergent dynamics of a generalized Lohe model on some class of Lie groups. J. Stat. Phys. 168, 171–207 (2017)
Ha, S.-Y., Ko, D., Ryoo, S.W.: On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds. J. Stat. Phys. 172, 1427–1478 (2018)
Kuramoto, Y.: International symposium on mathematical problems in mathematical physics. Lecture Notes Theor. Phys. 30, 420 (1975)
Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984)
Lohe, M.A.: Non-abelian Kuramoto model and synchronization. J. Phys. A: Math. Theor. 42, 395101 (2009)
Lohe, M.A.: Quantum synchronization over quantum networks. J. Phys. A: Math. Theor. 43, 465301 (2010)
Lohe, M.A.: Higher-dimensional generalizations of the Watanabe–Strogatz transform for vector models for synchronization. J. Phys. A: Math. Theor. 51, 225101 (2018)
Lohe, M.A.: Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization. J. Math. Phys. 60, 072701 (2019)
Olfati-Saber, R.: Swarms on sphere: a programmable swarm with synchronous behaviors like oscillator networks. In: IEEE 45th conference on decision and control (CDC) (2006), pp. 5060–5066
Peskin, C.S.: Mathematical Aspects of Heart Physiology. Courant Institute of Mathematical Sciences, New York (1975)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)
Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D 143, 1–20 (2000)
Verwoerd, M., Mason, O.: Global phase-locking in finite populations of phase-coupled oscillators. SIAM J. Appl. Dyn. Syst. 7, 134–160 (2008)
Verwoerd, M., Mason, O.: On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph. SIAM J. Appl. Dyn. Syst. 8, 417–453 (2009)
Watanabe, S., Strogatz, S.H.: Constants of motion for superconducting Josephson arrays. Phys. D 74, 197–253 (1994)
Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42 (1967)
Winfree, A.T.: The Geometry of Biological Time. Springer, New York (1980)
Acknowledgements
The work of S.-Y. Ha is supported by NRF-2020R1A2C3A01003881.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Alessandro Giuliani.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ha, SY., Park, H. On the Gradient Flow Formulation of the Lohe Matrix Model with High-Order Polynomial Couplings. J Stat Phys 184, 19 (2021). https://doi.org/10.1007/s10955-021-02804-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10955-021-02804-3