Abstract
We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott–Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the latter ones for the first time for identical oscillators). We show that twisted states can be stable starting from a certain critical value of the medium length, or on a length segment. The analytical results are confirmed with direct numerical simulations in finite ensembles.
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Pikovsky, A., Rosenblum, M., and Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences, New York: Cambridge Univ. Press, 2001.
Osipov, G., Kurths, J., and Zhou, C., Synchronization in Oscillatory Networks, Berlin: Springer, 2007.
Afraimovich, V. S., Nekorkin, V. I., Osipov, G.V., and Shalfeev, V. D., Stability, Structures and Chaos in Nonlinear Synchronization Networks, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 6, River Edge, N.J.: World Sci., 1994.
Pantaleone, J., Synchronization of Metronomes, Am. J. Phys., 2002, vol. 70, no. 10, pp. 992–1000.
Chhabria, S., Blaha, K. A., Rossa, F. D., and Sorrentino, F., Targeted Synchronization in an Externally Driven Population of Mechanical Oscillators, Chaos, 2018, vol. 28, no. 11, 111102, 6 pp.
Machowski, J., Bialek, J. W., and Bumby, J., Power System Dynamics: Stability and Control, 2nd ed., New Jersey: Wiley, 2012.
Menck, P. J., Heitzig, J., Kurths, J., and Schellnhuber, H. J., How Dead Ends Undermine Power Grid Stability, Nat. Commun., 2014, vol. 5, Art. 3969, 8 pp.
Ryu, S., Yu, W., and Stroud, D., Dynamics of an Underdamped Josephson-Junction Ladder, Phys. Rev. E, 1996, vol. 53, no. 3, pp. 2190–2195.
Zheng, Z., Hu, B., and Hu, B., Phase Slips and Phase Synchronization of Coupled Oscillators, Phys. Rev. Lett., 1998, vol. 81, no. 24, pp. 5318–5321.
Homma, S. and Takeno, S., A Coupled Base-Rotator Model for Structure and Dynamics of DNA: Local Fluctuations in Helical Twist Angles and Topological Solitons, Prog. Theor. Phys., 1984, vol. 72, no. 4, pp. 679–693.
Takeno, Sh. and Homma, Sh., Kinks and Breathers Associated with Collective Sugar Puckering in DNA, Progr. Theoret. Phys., 1987, vol. 77, no. 3, pp. 548–562.
Pikovsky, A. and Rosenblum, M., Dynamics of Globally Coupled Oscillators: Progress and Perspectives, Chaos, 2015, vol. 25, no. 9, 097616, 11 pp.
Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence, Springer Ser. Synergetics, vol. 19, Berlin: Springer, 1984.
Acebron, J., Bonilla, L., Vicente, C. P., Ritort, F., and Spigler, R., The Kuramoto Model: A Simple Paradigm for Synchronization Phenomena, Rev. Mod. Phys., 2005, vol. 77, no. 1, pp. 137–185.
Rodrigues, F. A., Peron, Th. K. D. M., Ji, P., and Kurths, J., The Kuramoto Model in Complex Networks, Phys. Rep., 2016, vol. 610, pp. 1–98.
Laing, C. R., The Dynamics of Chimera States in Heterogeneous Kuramoto Networks, Phys. D, 2009, vol. 238, no. 16, pp. 1569–1588.
Smirnov, L., Osipov, G., and Pikovsky, A., Chimera Patterns in the Kuramoto–Battogtokh Model, J. Phys. A, 2017, vol. 50, no. 8, 08LT01, 5 pp.
Bordyugov, G. A., Pikovsky, A. S., and Rosenblum, M. G., Self-Emerging and Turbulent Chimeras in Oscillator Chains, Phys. Rev. E., 2010, vol. 82, no. 3, 035205, 4 pp.
Girnyk, T., Hasler, M., and Maistrenko, Yu., Multistability of Twisted States in Non-Locally Coupled Kuramoto-Type Models, Chaos, 2012, vol. 22, no. 1, 013114, 10 pp.
Wiley, D. A., Strogatz, S. H., and Girvan, M., The Size of the Sync Basin, Chaos, 2006, vol. 16, no. 1, 015103, 8 pp.
Omel’chenko, O. E., Wolfrum, M., and Laing, C. R., Partially Coherent Twisted States in Arrays of Coupled Phase Oscillators, Chaos, 2014, vol. 24, no. 2, 023102, 9 pp.
Kuznetsov, S. P. and Mosekilde, E., Coupled Map Lattices with Complex Order Parameter, Phys. A, 2001, vol. 291, nos. 1–4, pp. 299–316.
Rosenblum, M. and Pikovsky, A., Self-Organized Quasiperiodicity in Oscillator Ensembles with Global Nonlinear Coupling, Phys. Rev. Lett., 2007, vol. 98, no. 6, 064101, 4 pp.
Pikovsky, A. and Rosenblum, M., Self-Organized Partially Synchronous Dynamics in Populations of Nonlinearly Coupled Oscillators, Phys. D, 2009, vol. 238, no. 1, pp. 27–37.
Ott, E. and Antonsen, Th. M., Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators, Chaos, 2008, vol. 18, no. 3, 037113, 6 pp.
Chowdhury, D. and Cross, M. C., Synchronization of Oscillators with Long-Range Power Law Interactions, Phys. Rev. E, 2010, vol. 82, no. 1, 016205, 12 pp.
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This work was supported by the RFBR (grant No. 19-52-12053) and the RSF (grant No. 19-12-00367).
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Bolotov, D., Bolotov, M., Smirnov, L. et al. Twisted States in a System of Nonlinearly Coupled Phase Oscillators. Regul. Chaot. Dyn. 24, 717–724 (2019). https://doi.org/10.1134/S1560354719060091
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DOI: https://doi.org/10.1134/S1560354719060091