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On the Completely Separable State for the Lohe Tensor Model

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Abstract

We study completely separable states of the Lohe tensor model and their asymptotic collective dynamics. Here, the completely separable state means that it is a tensor product of rank-1 tensors. For the generalized Lohe matrix model corresponding to the Lohe tensor model for rank-2 tensors with the same size, we observe that the component rank-1 tensors of the completely separable states satisfy the swarm double sphere model introduced in [Lohe in Physica D 412, 2020]. We also show that the swarm double sphere model can be represented as a gradient system with an analytic potential. Using this gradient flow formulation, we provide the swarm multisphere model on the product of multiple unit spheres with possibly different dimensions, and then we construct a completely separable state of the swarm multisphere model as a tensor product of rank-1 tensors which is a solution of the proposed swarm multisphere model. This concept of separability has been introduced in the theory of quantum information. Finally, we also provide a sufficient framework leading to the complete aggregation of completely separable states.

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References

  1. Albi, G., Bellomo, N., Fermo, L., Ha, S.-Y., Kim, J., Pareschi, L., Poyato, D., Soler, J.: Vehicular traffic, crowds, and swarms: from kinetic theory and multiscale methods to applications and research perspectives. Math. Models Methods Appl. Sci. 29, 1901–2005 (2019)

    Article  MathSciNet  Google Scholar 

  2. Barbǎlat, I.: Systèmes d’équations différentielles d’oscillations non linéaires. Rev. Math. Pures Appl. 4, 267–270 (1959)

    MathSciNet  MATH  Google Scholar 

  3. Buck, J., Buck, E.: Biology of synchronous flashing of fireflies. Nature 211, 562–564 (1966)

    Article  ADS  Google Scholar 

  4. Caponigro, M., Lai, A.C., Piccoli, B.: A nonlinear model of opinion formation on the sphere. Discret. Contin. Dyn. Syst. 35, 4241–4268 (2015)

    Article  MathSciNet  Google Scholar 

  5. Carrillo, J.A., Choi, Y.-P., Totzeck, C., Tse, O.: An analytical framework for consensus-based global optimization method. Math. Models Methods Appl. Sci. 28, 1037–1066 (2018)

    Article  MathSciNet  Google Scholar 

  6. Carrillo, J. A., Jin, S., Li, L., Zhu, Y.: A consensus-based global optimization method for high dimensional machine learning problems. To appear in ESAIM: Control, Optimisation and Calculus of Variations

  7. Chi, D., Choi, S.-H., Ha, S.-Y.: Emergent behaviors of a holonomic particle system on a sphere. J. Math. Phys. 55, 052703 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  8. Chandra, S., Girvan, M., Ott, E.: Continuous versus discontinuous transitions in the D-dimensional generalized Kuramoto model: Odd D is different. Phys. Rev. X 9, 011002 (2019)

    Google Scholar 

  9. Chandra, S., Ott, E.: Observing microscopic transitions from macroscopic bursts: instability-mediated resetting in the incoherent regime of the D-dimensional generalized Kuramoto model. Chaos 29, 033124 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  10. Choi, S.-H., Ha, S.-Y.: Complete entrainment of Lohe oscillators under attractive and repulsive couplings. SIAM J. Appl. Dyn. Syst. 13, 1417–1441 (2014)

    Article  MathSciNet  Google Scholar 

  11. Choi, Y.-P., Ha, S.-Y., Li, Z.: Emergent dynamics of the Cucker-Smale flocking model and its variants. Active particles. Vol. 1. Advances in theory, models, and applications, 299–331, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham (2017)

  12. Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52, 852–862 (2007)

    Article  MathSciNet  Google Scholar 

  13. Danino, T., Mondragon-Palomino, O., Tsimring, L., Hasty, J.: A synchronized quorum of genetic clocks. Nature 463, 326–330 (2010)

    Article  ADS  Google Scholar 

  14. DeVille, L.: Synchronization and stability for quantum Kuramoto. J. Stat. Phys. 174, 160–187 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  15. Frouvelle, A., Liu, J.-G.: Long-time dynamics for a simple aggregation equation on the sphere. Proc. Math. Stat. 282, 457–479 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Ha, S.-Y., Kim, D., Lee, J., No, S.-E.: Particle and kinetic models for swarming particles on a sphere and stability properties. J. Stat. Phys. 174, 622–655 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  17. Ha, S.-Y., Ko, D., Park, J., Zhang, X.: Collective synchronization of classical and quantum oscillators. EMS Surv. Math. Sci. 3, 209–267 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  18. Ha, S.-Y., Ko, D., Ryoo, S.: On the relaxation dynamics of Lohe oscillators on the Riemannian manifold. J. Stat. Phys. 172, 1427–1478 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  19. Ha, S.-Y., Park, H.: Emergent behaviors of Lohe tensor flocks. J. Stat. Phys. 178, 1268–1292 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  20. Ha, S.-Y., Park, H.: Emergent behaviors of the generalized Lohe matrix model. To appear in Discrete Contin. Dyn. Syst. B

  21. Hung, S.M., Givigi, S.N.: A q-learning approach to flocking with UAVs in a stochastic environment. IEEE Trans. Cybern. 47, 186–197 (2017)

    Article  Google Scholar 

  22. Kuramoto, Y.: 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, 420–422 (1975)

  23. Lohe, M.A.: Non-Abelian Kuramoto model and synchronization. J. Phys. A 42, 395101 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  24. Lohe, M.A.: High-dimensional generalizations of the Watanabe–Strogatz transform for vector models of synchronization. J. Phys. A 51, 225101 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  25. Lohe, M.A.: Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization. J. Math. Phys. 60, 072701 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  26. Lohe, M.A.: On the double sphere model of synchronization. Physica D. 412, 132642 (2020)

    Article  MathSciNet  Google Scholar 

  27. Markdahl, J., Gonçalves, J.:Global convergence properties of a consensus protocol on the \(n\)-sphere. Proc. of the 55th IEEE Conference on Decision and Control (CDC), pp. 2487–2492 (2016)

  28. Markdahl, J., Thunberg, J., Gonçalves, J.: Almost global consensus on the \(n\)-sphere. IEEE Trans. Autom. Control 63, 1664–1675 (2018)

    Article  MathSciNet  Google Scholar 

  29. Markdahl, J., Proverbio, D., Gonçalves, J.: Robust synchronization of heterogeneous robot swarms on the sphere. arXiv preprint: arXiv:2004.00296

  30. Nielsen, M.A.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  31. Olfati-Saber, R.: Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks. Proc. of the 45th IEEE Conference on Decision and Control, pp. 5060–5066 (2006)

  32. Perea, L., Elosegui, P., Gomez, G.: Extension of the Cucker–Smale control law to space flight formations. J. Guid. Control 32, 527–537 (2009)

    Article  ADS  Google Scholar 

  33. Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  34. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  36. Zhu, J.: Synchronization of Kuramoto model in a high-dimensional linear space. Phys. Lett. A 343, 2939–2943 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  37. Zhu, J., Zhu, J., Qian, C.: On equilibria and consensus of the Lohe model with identical oscillators. SIAM J. Appl. Dyns. Syst. 17, 1716–1741 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The work of S.-Y. Ha is supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881), 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).

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Authors and Affiliations

  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Republic of Korea

    Seung-Yeal Ha

  2. Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

    Seung-Yeal Ha

  3. School of Mathematics, Statistics and Data Science, Sungshin Women’s University, Seoul, 02844, Republic of Korea

    Dohyun Kim

  4. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Republic of Korea

    Hansol Park

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  1. Seung-Yeal Ha
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  2. Dohyun Kim
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  3. Hansol Park
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Correspondence to Dohyun Kim.

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Communicated by Irene Giardina.

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Ha, SY., Kim, D. & Park, H. On the Completely Separable State for the Lohe Tensor Model. J Stat Phys 183, 9 (2021). https://doi.org/10.1007/s10955-021-02750-0

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  • DOI: https://doi.org/10.1007/s10955-021-02750-0

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