Abstract

Many real-world systems of coupled agents exhibit directed interactions, meaning that the influence of an agent on another is not reciprocal. Furthermore, interactions usually do not have an identical amplitude and/or sign. To describe synchronization phenomena in such systems, we use a generalized Kuramoto model with oriented, weighted, and signed interactions. Taking a bottom-up approach, we investigate the simplest possible oriented networks, namely, acyclic oriented networks and oriented cycles. These two types of networks are fundamental building blocks from which many general oriented networks can be constructed. For acyclic, weighted, and signed networks, we are able to completely characterize synchronization properties through necessary and sufficient conditions, which we show are optimal. Additionally, we prove that if it exists, a stable synchronous state is unique. In oriented, weighted, and signed cycles with identical natural frequencies, we show that the system globally synchronizes and that the number of stable synchronous states is finite.

Keywords

  1. Kuramoto model
  2. synchronization
  3. directed graphs
  4. oriented graphs

MSC codes

  1. 34D06
  2. 37N35

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References

1.
Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in Proceedings of the International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys. 39, H. Araki, ed., Springer, Berlin, 1975, pp. 420--422.
2.
S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), pp. 1--20.
3.
J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Modern Phys., 77 (2005), 137.
4.
F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), pp. 1539--1564.
5.
G. B. Ermentrout, The behavior of rings of coupled oscillators, J. Math. Biol., 22 (1985), pp. 55--74.
6.
J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145.
7.
J. Buck, Synchronous rhythmic flashing of fireflies. II, Quart. Rev. Biol., 63 (1988), pp. 265--289.
8.
G. B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), pp. 571--585.
9.
Z. Lu, K. Klein-Carden͂a, S. Lee, T. M. Antonsen, M. Girvan, and E. Ott, Resynchronization of circadian oscillators and the east-west asymmetry of jet-lag, Chaos, 26 (2016), 094811.
10.
K. Wiesenfeld, P. Colet, and S. H. Strogatz, Synchronization transitions in a disordered Josephson series array, Phys. Rev. Lett., 76 (1996), p. 404.
11.
K. Wiesenfeld, P. Colet, and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E, 57 (1998), pp. 1563--1569.
12.
F. Dörfler, M. Chertkov, and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 2005--2010.
13.
E. Mallada and A. Tang, Synchronization of weakly coupled oscillators: Coupling, delay and topology, J. Phys. A, 46 (2013), 505101.
14.
H. Chiba, G. S. Medvedev, and M. S. Mizuhara, Bifurcations in the Kuramoto model on graphs, Chaos, 28 (2018), 073109.
15.
J. G. Restrepo, E. Ott, and B. R. Hunt, Synchronization in large directed networks of coupled phase oscillators, Chaos, 16 (2006), 015107.
16.
D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum, and J. K. Parrish, Oscillator models and collective motion, IEEE Control Syst., 27 (2007), pp. 89--105.
17.
R. Sepulchre, D. A. Paley, and N. E. Leonard, Stabilization of planar collective motion with limited communication, IEEE Trans. Automat. Control, 53 (2008), pp. 706--719.
18.
O. Mason and M. Verwoerd, Graph theory and networks in biology, IET Syst. Biol., 1 (2007), pp. 89--119.
19.
C. Börgers and N. Kopell, Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity, Neural Comput., 15 (2003), pp. 509--538.
20.
M. T. Schaub, N. O'Clery, Y. N. Billeh, J.-C. Delvenne, R. Lambiotte, and M. Barahona, Graph partitions and cluster synchronization in networks of oscillators, Chaos, 26 (2016), 094821.
21.
H. Hong and S. H. Strogatz, Kuramoto model of coupled oscillators with positive and negative coupling parameters: An example of conformist and contrarian oscillators, Phys. Rev. Lett., 106 (2011).
22.
O. Burylko, Competition and bifurcations in phase oscillator networks with positive and negative couplings, in Proceedings of the IEEE NDES, 2012, pp. 1--4.
23.
A. El Ati and E. Panteley, Asymptotic phase synchronization of Kuramoto model with weighted non-symmetric interconnections: A case study, in Proceedings of the 52nd IEEE CDC, 2013, pp. 1319--1324.
24.
A. El Ati and E. Panteley, Synchronization of phase oscillators with attractive and repulsive interconnections, in Proceedings of the 18th IEEE MMAR, 2013, pp. 22--27.
25.
P. S. Skardal, D. Taylor, and J. Sun, Optimal synchronization of directed complex networks, Chaos, 26 (2016), 094807.
26.
S.-Y. Ha and Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Commun. Math. Sci., 12 (2014), pp. 485--508.
27.
J. A. Rogge and D. Aeyels, Stability of phase locking in a ring of unidirectionally coupled oscillators, J. Phys. A, 37 (2004), pp. 11135--11148.
28.
S.-Y. Ha and M.-J. Kang, On the basin of attractors for the unidirectionally coupled Kuramoto model in a ring, SIAM J. Appl. Math., 72 (2012), pp. 1549--1574.
29.
R. Potrie and P. Monzón, Local implications of almost global stability, Dyn. Syst., 24 (2009), pp. 109--115.
30.
K. Mischaikow, H. Smith, and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995).
31.
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1976.
32.
H. K. Khalil, Nonlinear Systems, 3rd ed., Prentice-Hall, Englewood Cliffs, NJ, 2002.
33.
R. Delabays, T. Coletta, and Ph. Jacquod, Multistability of phase-locking and topological winding numbers in locally coupled Kuramoto models on single-loop networks, J. Math. Phys., 57 (2016), 032701.
34.
J. P. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method: With Applications, Math. Sci. Eng. 4, Academic Press, New York, 1961.
35.
D. Manik, M. Timme, and D. Witthaut, Cycle flows and multistability in oscillatory networks, Chaos, 27 (2017), 083123.
36.
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1986.
37.
L. G. Khazin and E. E. Shnol, Stability of Critical Equilibrium States, Manchester University Press, Manchester, UK, 1991.
38.
N. Janssens and A. Kamagate, Loop flows in a ring AC power system, Int. J. Elect. Power Energy Syst., 25 (2003), pp. 591--597.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 458 - 480
ISSN (online): 1536-0040

History

Submitted: 31 July 2018
Accepted: 16 January 2019
Published online: 28 February 2019

Keywords

  1. Kuramoto model
  2. synchronization
  3. directed graphs
  4. oriented graphs

MSC codes

  1. 34D06
  2. 37N35

Authors

Affiliations

Funding Information

Eidgenössische Technische Hochschule Zürich https://doi.org/10.13039/501100003006 : PYAPP2 154275

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