Spiral wave chimeras induced by heterogeneity in phase lags and time delays. (English) Zbl 1512.35060
Summary: A spiral wave chimera is a remarkable spatiotemporal pattern in a two-dimensional array of oscillators, for which the coherent spiral arms coexist with incoherent cores. So far the spiral wave chimeras have been known to occur in nonlocally coupled oscillators where the coupling strength between oscillators varies with the distance between them. Here we report on spiral wave chimeras for globally coupled phase oscillators with heterogeneous phase lags on the sphere. On the basis of Ott-Antonsen theory, we reduce our model to a low-dimensional system and present stability diagrams for different stationary states of the reduced system. We demonstrate the existence of spiral wave chimeras for the globally coupled phase oscillators with space-dependent interaction delays on the sphere, which are extended to appear also in the Stuart-Landau system of amplitude-phase oscillators. Chimeric behavior due to the heterogeneity in phase lags or time delays is peculiar to two-dimensional arrays of oscillators, which exhibits a self-emerging state in a wide parameter region. As an essential driving mechanism for the emergence of spiral chimeras, the space-dependent feature of interaction delays is omnipresent in nature and engineering systems, and we anticipate that our model and the related spiral chimera patterns will have widespread practical applications in the biological oscillatory networks.
MSC:
| 35B36 | Pattern formations in context of PDEs |
| 35R09 | Integro-partial differential equations |
| 35R10 | Partial functional-differential equations |
Keywords:
spiral wave Chimera; coupled oscillators; nonlocal coupling; Ott-Antonsen reduction; space-dependent delaysReferences:
| [1] | Winfree, A. T., The Geometry of Biological Time (2001), Springer: Springer New York · Zbl 1014.92001 |
| [2] | Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences (2001), Cambridge University Press · Zbl 0993.37002 |
| [3] | Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence (1984), Springer-Verlag · Zbl 0558.76051 |
| [4] | Panfilov, A. V., Spiral breakup as a model of ventricular fibrillation, Chaos, 8, 57-64 (1998) · Zbl 1069.92509 |
| [5] | Davidenko, J. M.; Pertsov, A. V.; Salomonsz, R.; Baxter, W.; Jalife, J., Stationary and drifting spiral waves of excitation in isolated cardiac muscle, Nature, 355, 349-351 (1992) |
| [6] | Winfree, A. T., Electrical turbulence in three-dimensional heart muscle, Science, 266, 1003-1006 (1994) |
| [7] | Cherry, E. M.; Fenton, F. H., Visualization of spiral and scroll waves in simulated and experimental cardiac tissue, New J. Phys., 10, Article 125016 pp. (2008) |
| [8] | Huang, X.; Troy, W. C.; Yang, Q.; Ma, H.; Laing, C. R.; Schiff, S. J.; Wu, J.-Y., Spiral waves in disinhibited mammalian Neo cortex, J. Neurosci., 24, 9897-9902 (2004) |
| [9] | Kuramoto, Y.; Battogtokh, D., Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst., 5, 380-385 (2002) |
| [10] | Abrams, D. M.; Strogatz, S. H., Chimera states for coupled oscillators, Phys. Rev. Lett., 93, Article 174102 pp. (2004) |
| [11] | Abrams, D. M.; Strogatz, S. H., Chimera states in a ring of nonlocally coupled oscillators, Int. J. Bifurcation Chaos, 16, 21-37 (2006) · Zbl 1101.37319 |
| [12] | Panaggio, M. J.; Abrams, D. M., Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators, Nonlinearity, 28, R67-R87 (2015) · Zbl 1392.34036 |
| [13] | Schöll, E., Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics, Eur. Phys. J. Spec. Top., 225, 891-919 (2016) |
| [14] | Omel’chenko, O. E., The mathematics behind chimera states, Nonlinearity, 31, R121-R164 (2018) · Zbl 1395.34045 |
| [15] | Shima, S.; Kuramoto, Y., Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators, Phys. Rev. E, 69, Article 036213 pp. (2004) |
| [16] | Laing, C., The dynamics of chimera states in heterogeneous Kuramoto networks, Physica D, 238, 1569-1588 (2009) · Zbl 1185.34042 |
| [17] | Martens, E. A.; Laing, C. R.; Strogatz, S. H., Solvable model of spiral wave chimeras, Phys. Rev. Lett., 104, Article 044101 pp. (2010) |
| [18] | Omel’chenko, O. E.; Wolfrum, M.; Yanchuk, S.; Maistrenko, Y. L.; Sudakov, O., Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally coupled phase oscillators, Phys. Rev. E, 85, Article 036210 pp. (2012) |
| [19] | Gu, C.; St-Yves, G.; Davidsen, J., Spiral wave chimeras in complex oscillatory and chaotic systems, Phys. Rev. Lett., 111, Article 134101 pp. (2013) |
| [20] | Panaggio, M. J.; Abrams, D. M., Chimera states on the surface of a sphere, Phys. Rev. E, 91, Article 022909 pp. (2015) |
| [21] | Xie, J.; Knobloch, E.; Kao, H.-C., Twisted chimera states and multicore spiral chimera states on a two-dimensional torus, Phys. Rev. E, 92, Article 042921 pp. (2015) |
| [22] | Li, B.-W.; Dierckx, H., Spiral wave chimeras in locally coupled oscillator systems, Phys. Rev. E, 93, Article 020202 pp. (2016) |
| [23] | Schmidt, A.; Kasimatis, T.; Hizanidis, J.; Provata, A.; Hövel, P., Chimera patterns in two-dimensional networks of coupled neurons, Phys. Rev. E, 95, Article 032224 pp. (2017) |
| [24] | Laing, C., Chimeras in two-dimensional domains: Heterogeneity and the continuum limit, SIAM J. Appl. Dyn. Syst., 16, 974-1014 (2017) · Zbl 1392.34035 |
| [25] | Omel’chenko, O. E.; Wolfrum, M.; Knobloch, E., Stability of spiral chimera states on a torus, SIAM J. Appl. Dyn. Syst., 17, 97-127 (2018) · Zbl 1386.37047 |
| [26] | Kim, R.-S.; Choe, C.-U., Symmetry-broken states on a spherical surface of coupled oscillators: From modulated coherence to spot and spiral chimeras, Phys. Rev. E, 98, Article 042207 pp. (2018) |
| [27] | Totz, J. F.; Rode, J.; Tinsley, M. R.; Showalter, K.; Engel, H., Spiral wave chimera states in large populations of coupled chemical oscillators, Nat. Phys., 14, 282-285 (2018) |
| [28] | Davidsen, J., Symmetry-breaking spirals, Nat. Phys. (2017) |
| [29] | Fiedler, B.; Flunkert, V.; Georgi, M.; Hövel, P.; Schöll, E., Refuting the odd number limitation of time-delayed feedback control, Phys. Rev. Lett., 98, Article 114101 pp. (2007) |
| [30] | Just, W.; Fiedler, B.; Flunkert, V.; Georgi, M.; Hövel, P.; Schöll, E., Beyond odd number limitation: a bifurcation analysis of time-delayed feedback control, Phys. Rev. E, 76, Article 026210 pp. (2007) |
| [31] | Schikora, S.; Wünsche, H. J.; Henneberger, F., Odd-number theorem: optical feedback control at a subcritical Hopf bifurcation in a semiconductor laser, Phys. Rev. E, 83, Article 026203 pp. (2011) |
| [32] | Choe, C.-U.; Jang, H.; Flunkert, V.; Dahms, T.; Hövel, P.; Schöll, E., Stabilization of periodic orbits near a subcritical Hopf bifurcation in delay-coupled networks, Dyn. Syst., 28, 15-33 (2013) · Zbl 1277.34109 |
| [33] | Pyragas, K.; Pyragiene, T., Coupling design for a long-term anticipating synchronization of chaos, Phys. Rev. E, 78, Article 046217 pp. (2008) |
| [34] | Pyragas, K.; Pyragiene, T., Extending anticipation horizon of chaos synchronization schemes with time-delay coupling, Philos. Trans. R. Soc. A, 368, 305-317 (2010) · Zbl 1204.34103 |
| [35] | Peil, M.; Heil, T.; Fischer, I.; Elsässer, W., Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario, Phys. Rev. Lett., 88, Article 174101 pp. (2002) |
| [36] | Flunkert, V.; Schöll, E., Chaos synchronization in networks of delay-coupled lasers: role of the coupling phases, New J. Phys., 14, Article 033039 pp. (2012) |
| [37] | Choe, C.-U.; Dahms, T.; Hövel, P.; Schöll, E., Controlling synchrony by delay coupling in networks: from in-phase to splay cluster states, Phys. Rev. E, 81, Article 025205 pp. (2010), (R) |
| [38] | Selivanov, A. A.; Lehnert, J.; Dahms, T.; Hövel, P.; Fradkov, A. L.; Schöll, E., Adaptive synchronization in delay-coupled networks of Stuart-Landau oscillators, Phys. Rev. E, 85, Article 016201 pp. (2012) · Zbl 1260.34107 |
| [39] | Choe, C.-U.; Ri, J.-S.; Kim, R.-S., Incoherent chimera and glassy states in coupled oscillators with frustrated interactions, Phys. Rev. E, 94, Article 032205 pp. (2016) |
| [40] | Choe, C.-U.; Kim, R.-S.; Ri, J.-S., Chimera and modulated drift states in a ring of nonlocally coupled oscillators with heterogeneous phase lags, Phys. Rev. E, 96, Article 032224 pp. (2017) |
| [41] | Kim, R.-S.; Choe, C.-U., Chimera state on a spherical surface of nonlocally coupled oscillators with heterogeneous phase lags, Chaos, 29, Article 023101 pp. (2019) · Zbl 1409.34049 |
| [42] | Crook, S. M.; Ermentrout, G. B.; Vanier, M. C.; Bower, J. M., The role of axonal delay in the synchronization of networks of coupled cortical oscillators, J. Comput. Neurosci., 4, 161-172 (1997) · Zbl 0893.92004 |
| [43] | Lohe, M. A., Synchronization control in networks with uniform and distributed phase lag, Automatica, 54, 114-123 (2015) · Zbl 1318.93040 |
| [44] | Dörfler, F.; Bullo, F., Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators, SIAM J. Control Optim., 50, 1616-1642 (2012) · Zbl 1264.34105 |
| [45] | Motter, A. E.; Myers, S. A.; Anghel, M.; Nishikawa, T., Spontaneous synchrony in power-grid networks, Nat. Phys., 9, 191-197 (2013) |
| [46] | Tyrrell, A.; Auer, G.; Bettstetter, C., Emergent slot synchronization in wireless networks, IEEE Trans. Mobile Comput., 9, 719-732 (2010) |
| [47] | Venkov, N. A.; Coombes, S.; Matthews, P. C., Dynamic instabilities in scalar neural field equations with space-dependent delays, Physica D, 232, 1-15 (2007) · Zbl 1127.45002 |
| [48] | Song, Q.; Wang, Z., Neural networks with discrete and distributed time-varying delays: A general stability analysis, Chaos Solitons Fractals, 37, 1538-1547 (2008) · Zbl 1142.34380 |
| [49] | Coombes, S., Large-scale neural dynamics: Simple and complex, NeuroImage, 52, 731-739 (2010) |
| [50] | Visser, S.; Nicks, R.; Faugeras, O.; Coombes, S., Standing and travelling waves in a spherical brain model: The Nunez model revisited, Physica D (2017) · Zbl 1376.92014 |
| [51] | Ott, E.; Antonsen, T. M., Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18, Article 037113 pp. (2008) · Zbl 1309.34058 |
| [52] | Ott, E.; Antonsen, T. M., Long time evolution of phase oscillator systems, Chaos, 19, Article 023117 pp. (2009) · Zbl 1309.34059 |
| [53] | Ott, E.; Hunt, B. R.; Antonsen, T. M., Comment on “Long time evolution of phase oscillator systems” [Chaos 19, 023117 (2009)], Chaos, 21, Article 025112 pp. (2011) · Zbl 1317.34064 |
| [54] | Pikovsky, A.; Rosenblum, M., Partially integrable dynamics of hierarchical populations of coupled oscillators, Phys. Rev. Lett., 101, Article 264103 pp. (2008) |
| [55] | Pikovsky, A.; Rosenblum, M., Dynamics of heterogeneous oscillator ensembles in terms of collective variables, Physica D, 240, 872-881 (2011) · Zbl 1233.37014 |
| [56] | Watanabe, S.; Strogatz, S. H., Constants of motion for superconducting josephson arrays, Physica D, 74, 197-253 (1994) · Zbl 0812.34043 |
| [57] | Sethia, G. C.; Sen, A.; Atay, F. M., Clustered chimera states in delay-coupled oscillator systems, Phys. Rev. Lett., 100, Article 144102 pp. (2008) |
| [58] | Calhoun, D. A.; Helzel, C.; LeVeque, R. J., Logically rectangular finite volume grids and methods for circular and spherical domains, SIAM Rev., 50, 723-752 (2008) · Zbl 1155.65061 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
