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The dynamics of chimera states in heterogeneous Kuramoto networks. (English) Zbl 1185.34042
The paper presents a number of analytical results concerning emergence and stability of chimera states in networks of coupled phase oscillators. The chimera states have previously been discovered as a surprise in networks of symmetrically coupled identical oscillators, where the population spontaneously splits into synchronized and desynchronized subpopulations. The present paper reports on the existence and stability of the chimera states for nonidentical oscillators if a heterogeneity of the natural frequencies is introduced. It is shown that a power-law ansatz for the probability density function recently discovered by Ott and Antonsen provides an effective tool for the investigation of stationary states of the evolutionary equation of the network. The paper addresses a variety of models including one- and two-dimensional oscillator arrays with non-local coupling, all-to-all coupled networks with inhomogeneous coupling strengths as well as delay-coupled oscillator ensembles. Bifurcation analysis of the stationary chimera states demonstrates that these states are robust with respect to small heterogeneity in the networks. The obtained analytical results are illustrated by numerical simulations.

MSC:
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
Software:
AUTO; HomCont
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References:
[1] Strogatz, S., Sync: the emerging science of spontaneous order, (2003), Hyperion
[2] Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization, (2001), Cambridge University Press
[3] Winfree, A.T., The geometry of biological time, (2001), Springer · Zbl 0856.92002
[4] Acebrón, J.A.; Bonilla, L.L.; Pérez Vicente, C.J.; Ritort, F.; Spigler, R., The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. modern phys., 77, 137-185, (2005)
[5] Kuramoto, Y., Chemical oscillations, waves, and turbulence, (1984), Springer-Verlag · Zbl 0558.76051
[6] Strogatz, S.H., From Kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143, 1-20, (2000) · Zbl 0983.34022
[7] Abrams, D.M.; Strogatz, S.H., Chimera states in a ring of nonlocally coupled oscillators, Int. J. bifur. chaos, 16, 21-37, (2006) · Zbl 1101.37319
[8] Abrams, D.M.; Mirollo, R.; Strogatz, S.H.; Wiley, D.A., Solvable model for Chimera states of coupled oscillators, Phys. rev. lett., 101, 084103, (2008)
[9] Sethia, G.C.; Sen, A.; Atay, F.M., Clustered Chimera states in delay-coupled oscillator systems, Phys. rev. lett., 100, 144102, (2008)
[10] Abrams, D.M.; Strogatz, S.H., Chimera states for coupled oscillators, Phys. rev. lett., 93, 174102, (2004)
[11] Omel’chenko, O.E.; Maistrenko, Y.L.; Tass, P.A., Chimera states: the natural link between coherence and incoherence, Phys. rev. lett., 100, 044105, (2008)
[12] Laing, C.R., Chimera states in heterogeneous networks, Chaos, 19, 013113, (2009) · Zbl 1311.34080
[13] Ko, T.W.; Ermentrout, G.B., Partially locked states in coupled oscillators due to inhomogeneous coupling, Phys. rev. E, 78, 1, 016203, (2008)
[14] Kuramoto, Y.; Battogtokh, D., Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear phenom. complex syst., 5, 380-385, (2002)
[15] Shima, S.; Kuramoto, Y., Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators, Phys. rev. E, 69, 3, 036213, (2004)
[16] Ott, E.; Antonsen, T.M., Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18, 037113, (2008) · Zbl 1309.34058
[17] Pikovsky, A.; Rosenblum, M., Partially integrable dynamics of hierarchical populations of coupled oscillators, Phys. rev. lett., 101, 264103, (2008)
[18] Watanabe, S.; Strogatz, S.H., Constants of motion for superconducting Josephson arrays, Physica D, 74, 197-253, (1994) · Zbl 0812.34043
[19] Martens, E.A.; Barreto, E.; Strogatz, S.H.; Ott, E.; So, P.; Antonsen, T.M., Exact results for the Kuramoto model with a bimodal frequency distribution, Phys. rev. E, 79, 026204, (2009)
[20] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer · Zbl 0515.34001
[21] Kuznetsov, Y.A., Elements of applied bifurcation theory, (2004), Springer · Zbl 1082.37002
[22] E. Doedel, R.C. Paffenroth, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X. Wang, AUTO 2000: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), Concordia University, Canada, ftp.cs.concordia.ca/pub/doedel/auto
[23] Laing, C.; Coombes, S., The importance of different timings of excitatory and inhibitory pathways in neural field models, Network: comput. neural syst., 17, 2, 151-172, (2006)
[24] Bär, M.; Bangia, A.K.; Kevrekidis, I.G., Bifurcation and stability analysis of rotating chemical spirals in circular domains: boundary-induced meandering and stabilization, Phys. rev. E, 67, 5, 056126, (2003)
[25] Barkley, D., Linear stability analysis of rotating spiral waves in excitable media, Phys. rev. lett., 68, 13, 2090-2093, (1992)
[26] Barkley, D., Euclidean symmetry and the dynamics of rotating spiral waves, Phys. rev. lett., 72, 1, 164-167, (1994)
[27] Kim, P.J.; Ko, T.W.; Jeong, H.; Moon, H.T., Pattern formation in a two-dimensional array of oscillators with phase-shifted coupling, Phys. rev. E, 70, 6, 065201, (2004)
[28] Bordyugov, G.; Engel, H., Continuation of spiral waves, Physica D, 228, 1, 49-58, (2007) · Zbl 1114.35099
[29] Laing, C.R., Spiral waves in nonlocal equations, SIAM J. appl. dynamical systems, 4, 3, 588-606, (2005) · Zbl 1090.37056
[30] Marvel, S.A.; Strogatz, S.H., Invariant submanifold for series arrays of Josephson junctions, Chaos, 19, 013132, (2009) · Zbl 1311.37015
[31] Childs, L.M.; Strogatz, S.H., Stability diagram for the forced Kuramoto model, Chaos, 18, 043128, (2008) · Zbl 1309.34088
[32] E. Ott, T.M. Antonsen, Long time evolution of phase oscillator systems, Arxiv preprint arXiv:0902.2773v1, 2009 · Zbl 1309.34059
[33] S.A. Marvel, R.E. Mirollo, S.H. Strogatz, Sinusoidally coupled phase oscillators evolve by Möbius group action, Arxiv preprint, arXiv:0904.1680v1, 2009 · Zbl 1311.34082
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