Time-delayed interactions and synchronization of identical Lohe oscillators

Choi, Sun-Ho; Ha, Seung-Yeal

doi:10.1090/qam/1417

Authors: Sun-Ho Choi and Seung-Yeal Ha
Journal: Quart. Appl. Math. 74 (2016), 297-319
MSC (2010): Primary 92D25, 74A25, 76N10
DOI: https://doi.org/10.1090/qam/1417
Published electronically: March 18, 2016
MathSciNet review: 3505605
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Abstract: We study time-delay effects on the synchronous dynamics of identical Lohe oscillators on the unit sphere. Time delays in the interactions between Lohe oscillators are induced by the finite propagation speed of information or communication, and generate some oscillatory phenomena in the initial time-layer near the initial time. From the viewpoint of synchronization, we provide a sufficient framework for the complete positional synchronization of Lohe oscillators in terms of their initial configuration, time delay, and coupling strength. As long as the time delay is sufficiently small, there is no qualitative effect on synchronization.

References

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.
J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211 (1966), 562 – 564.
Dongpyo Chi, Sun-Ho Choi, and Seung-Yeal Ha, Emergent behaviors of a holonomic particle system on a sphere, J. Math. Phys. 55 (2014), no. 5, 052703, 18. MR 3390625, DOI 10.1063/1.4878117
Sun-Ho Choi and Seung-Yeal Ha, Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM J. Appl. Dyn. Syst. 13 (2014), no. 4, 1417–1441. MR 3267148, DOI 10.1137/140961699
M. Y. Choi, H. J. Kim, D. Kim, and H. Hong, Synchronization in a system of globally coupled oscillators with time delay, Phys. Review E. 61, 371 – 381 (2000).
G. Bard Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol. 22 (1985), no. 1, 1–9. MR 802731, DOI 10.1007/BF00276542
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721
H. Kook, S.-G. Lee, D.-U. Hwang, and S. K. Han, Synchronization of a neuronal oscillator network with multiple connections of time delays, J. Korean Phys. Society 50, 341 – 345 (2007).
Yang Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, Inc., Boston, MA, 1993. MR 1218880
Yoshiki Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975) Springer, Berlin, 1975, pp. 420–422. Lecture Notes in Phys., 39. MR 0676492
J. J. Levin and J. A. Nohel, On a nonlinear delay equation, J. Math. Anal. Appl. 8 (1964), 31–44. MR 163142, DOI 10.1016/0022-247X(64)90080-0
M. A. Lohe, Non-abelian Kuramoto models and synchronization, J. Phys. A 42 (2009), no. 39, 395101, 25. MR 2539317, DOI 10.1088/1751-8113/42/39/395101
M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A 43 (2010), no. 46, 465301, 20. MR 2735225, DOI 10.1088/1751-8113/43/46/465301
R. Olfati-Saber, Swarms on Sphere: A Programmable Swarm with Synchronous Behaviors like Oscillator Networks, Decision and Control, 2006 (45th IEEE Conference on).
Reza Olfati-Saber and Richard M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Control 49 (2004), no. 9, 1520–1533. MR 2086916, DOI 10.1109/TAC.2004.834113
Antonis Papachristodoulou, Ali Jadbabaie, and Ulrich Münz, Effects of delay in multi-agent consensus and oscillator synchronization, IEEE Trans. Automat. Control 55 (2010), no. 6, 1471–1477. MR 2668959, DOI 10.1109/TAC.2010.2044274
Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. A universal concept in nonlinear sciences. MR 1869044, DOI 10.1017/CBO9780511755743
D. V. Ramana Reddy, A. Sen, and G. L. Johnston, Time delay effects on coupled limit cycle oscillators at Hopf bifurcation, Phys. D 129 (1999), no. 1-2, 15–34. MR 1690281, DOI 10.1016/S0167-2789(99)00004-4
Gerd S. Schmidt, Antonis Papachristodoulou, Ulrich Münz, and Frank Allgöwer, Frequency synchronization and phase agreement in Kuramoto oscillator networks with delays, Automatica J. IFAC 48 (2012), no. 12, 3008–3017. MR 2995676, DOI 10.1016/j.automatica.2012.08.013
Marshall Slemrod, The flip-flop circuit as a neutral equation, Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972) Academic Press, New York, 1972, pp. 387–392. MR 0397117
Marshall Slemrod and E. F. Infante, Asymptotic stability criteria for linear systems of difference-differential equations of neutral type and their discrete analogues, J. Math. Anal. Appl. 38 (1972), 399–415. MR 306678, DOI 10.1016/0022-247X(72)90098-4
Hal Smith, An introduction to delay differential equations with applications to the life sciences, Texts in Applied Mathematics, vol. 57, Springer, New York, 2011. MR 2724792, DOI 10.1007/978-1-4419-7646-8
Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D 143 (2000), no. 1-4, 1–20. Bifurcations, patterns and symmetry. MR 1783382, DOI 10.1016/S0167-2789(00)00094-4
Arthur T. Winfree, The geometry of biological time, Biomathematics, vol. 8, Springer-Verlag, Berlin-New York, 1980. MR 572965
A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15 – 42 (1967).
Jiandong Zhu, Synchronization of Kuramoto model in a high-dimensional linear space, Phys. Lett. A 377 (2013), no. 41, 2939–2943. MR 3108680, DOI 10.1016/j.physleta.2013.09.010