Letter | Published:

Spiral wave chimera states in large populations of coupled chemical oscillators

Nature Physicsvolume 14pages282285 (2018) | Download Citation

Abstract

The coexistence of coherent and incoherent dynamics in a population of identically coupled oscillators is known as a chimera state1,2. Discovered in 20023, this counterintuitive dynamical behaviour has inspired extensive theoretical and experimental activity4,5,6,7,8,9,10,11,12,13,14,15. The spiral wave chimera is a particularly remarkable chimera state, in which an ordered spiral wave rotates around a core consisting of asynchronous oscillators. Spiral wave chimeras were theoretically predicted in 200416 and numerically studied in a variety of systems17,18,19,20,21,22,23. Here, we report their experimental verification using large populations of nonlocally coupled Belousov–Zhabotinsky chemical oscillators10,18 in a two-dimensional array. We characterize previously unreported spatiotemporal dynamics, including erratic motion of the asynchronous spiral core, growth and splitting of the cores, as well as the transition from the chimera state to disordered behaviour. Spiral wave chimeras are likely to occur in other systems with long-range interactions, such as cortical tissues24, cilia carpets25, SQUID metamaterials26 and arrays of optomechanical oscillators9.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

  1. 1.

    Abrams, D. M. & Strogatz, S. H. Chimera states for coupled oscillators. Phys. Rev. Lett. 93, 174102 (2004).

  2. 2.

    Panaggio, M. J. & Abrams, D. M. Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28, R67 (2015).

  3. 3.

    Kuramoto, Y. & Battogtokh, D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlin. Phenom. Complex Syst. 5, 380–385 (2002).

  4. 4.

    Zakharova, A., Kapeller, M. & Schöll, E. Chimera death: symmetry breaking in dynamical networks. Phys. Rev. Lett. 112, 154101 (2014).

  5. 5.

    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).

  6. 6.

    Wolfrum, M. & Omel’chenko, O. E. Chimera states are chaotic transients. Phys. Rev. E 84, 015201 (2011).

  7. 7.

    Sethia, G. C., Sen, A. & Atay, F. M. Clustered chimera states in delay-coupled oscillator systems. Phys. Rev. Lett. 100, 144102 (2008).

  8. 8.

    Schmidt, L., Schönleber, K., Krischer, K. & García-Morales, V. Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling. Chaos 24, 013102 (2014).

  9. 9.

    Hagerstrom, A. M. et al. Experimental observation of chimeras in coupled-map lattices. Nat. Phys. 8, 658–661 (2012).

  10. 10.

    Tinsley, M. R., Nkomo, S. & Showalter, K. Chimera and phase-cluster states in populations of coupled chemical oscillators. Nat. Phys. 8, 662–665 (2012).

  11. 11.

    Martens, E. A., Thutupalli, S., Fourrière, A. & Hallatschek, O. Chimera states in mechanical oscillator networks. Proc. Natl Acad. Sci. USA 110, 10563–10567 (2013).

  12. 12.

    Wojewoda, J., Czolczynski, K., Maistrenko, Y. & Kapitaniak, T. The smallest chimera state for coupled pendula. Sci. Rep. 6, 34329 (2016).

  13. 13.

    Wickramasinghe, M. & Kiss, I. Z. Spatially organized dynamical states in chemical oscillator networks: synchronization, dynamical differentiation, and chimera patterns. PLoS ONE 8, e80586 (2013).

  14. 14.

    Larger, L., Penkovsky, B. & Maistrenko, Y. Laser chimeras as a paradigm for multistable patterns in complex systems. Nat. Commun. 6, 7752 (2015).

  15. 15.

    Rosin, D. P., Rontani, D., Haynes, N. D., Schöll, E. & Gauthier, D. J. Transient scaling and resurgence of chimera states in networks of Boolean phase oscillators. Phys. Rev. E 90, 030902 (2014).

  16. 16.

    Shima, S. & Kuramoto, Y. Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. Phys. Rev. E 69, 036213 (2004).

  17. 17.

    Martens, E. A., Laing, C. R. & Strogatz, S. H. Solvable model of spiral wave chimeras. Phys. Rev. Lett. 104, 044101 (2010).

  18. 18.

    Nkomo, S., Tinsley, M. R. & Showalter, K. Chimera states in populations of nonlocally coupled chemical oscillators. Phys. Rev. Lett. 110, 244102 (2013).

  19. 19.

    Gu, C., St-Yves, G. & Davidsen, J. Spiral wave chimeras in complex oscillatory and chaotic systems. Phys. Rev. Lett. 111, 134101 (2013).

  20. 20.

    Tang, X. et al. Novel type of chimera spiral waves arising from decoupling of a diffusible component. J. Chem. Phys. 141, 024110 (2014).

  21. 21.

    Xie, J. B., Knobloch, E. & Kao, H. C. Twisted chimera states and multicore spiral chimera states on a two-dimensional torus. Phys. Rev. E 92, 042921 (2015).

  22. 22.

    Lau, H. W. & Davidsen, J. Linked and knotted chimera filaments in oscillatory systems. Phys. Rev. E 94, 010204 (2016).

  23. 23.

    Laing, C. R. Chimeras in two-dimensional domains: heterogeneity and the continuum limit. SIAM J. Appl. Dyn. Sys. 16, 974–1014 (2017).

  24. 24.

    Huang, X. et al. Spiral waves in disinhibited mammalian neocortex. J. Neurosci. 24, 9897–9902 (2004).

  25. 25.

    Uchida, N. & Golestanian, R. Synchronization and collective dynamics in a carpet of microfluidic rotors. Phys. Rev. Lett. 104, 178103 (2010).

  26. 26.

    Lazarides, N., Neofotistos, G. & Tsironis, G. P. Chimeras in SQUID metamaterials. Phys. Rev. B 91, 054303 (2015).

  27. 27.

    Winfree, A. T. The Geometry of Biological Time (Springer, New York, 2001).

  28. 28.

    Zhabotinsky, A. M., Buchholtz, F., Kiyatkin, A. B. & Epstein, I. R. Oscillations and waves in metal-ion-catalyzed bromate oscillating reactions in highly oxidized states. J. Phys. Chem. 97, 7578–7584 (1993).

  29. 29.

    Davidsen, J., Glass, L. & Kapral, R. Topological constraints on spiral wave dynamics in spherical geometries with inhomogeneous excitability. Phys. Rev. E 70, 056203 (2004).

  30. 30.

    Canavier, C. C. & Achuthan, S. Pulse coupled oscillators and the phase resetting curve. Math. Biosci. 226, 77–96 (2010).

  31. 31.

    Taylor, A. F. et al. Clusters and switchers in globally coupled photochemical oscillators. Phys. Rev. Lett. 100, 214101 (2008).

  32. 32.

    Totz, J. F. et al. Phase-lag synchronization in networks of coupled chemical oscillators. Phys. Rev. E 92, 022819 (2015).

  33. 33.

    Nkomo, S., Tinsley, M. R. & Showalter, K. Chimera and chimera-like states in populations of nonlocally coupled homogeneous and heterogeneous chemical oscillators. Chaos 26, 094826 (2016).

Download references

Acknowledgements

The authors thank J. Sixt and F. Sielaff from TU Berlin Physics Department’s precision mechanical workshop for preparing the acrylic glass plates with micrometre-sized cavities that hold the micro-oscillators and U. Künkel for assistance in the laboratory. This work was supported by the Deutsche Forschungsgemeinschaft (grants GRK 1558 and SFB 910 to J.F.T., J.R. and H.E.), the National Science Foundation (grant CHE-1565665 to K.S. and M.R.T.) and the Alexander von Humboldt-Stiftung (to K.S.).

Author information

Affiliations

  1. Institut für Theoretische Physik, EW 7-1, TU Berlin, Hardenbergstr. 36, 10623, Berlin, Germany

    • Jan Frederik Totz
    • , Julian Rode
    •  & Harald Engel
  2. C. Eugene Bennett Department of Chemistry, West Virginia University, Morgantown, WV, 26506-6045, USA

    • Mark R. Tinsley
    •  & Kenneth Showalter

Authors

  1. Search for Jan Frederik Totz in:

  2. Search for Julian Rode in:

  3. Search for Mark R. Tinsley in:

  4. Search for Kenneth Showalter in:

  5. Search for Harald Engel in:

Contributions

J.F.T. and J.R. built and programmed the set-up and performed experiments. J.F.T., K.S. and H.E. designed the study and wrote the paper. The simulations were carried out by J.F.T. and J.R., except for those shown in Fig. 3b–i, which were done by M.R.T. All authors discussed the results and commented on the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to Jan Frederik Totz or Kenneth Showalter or Harald Engel.

Electronic supplementary material

About this article

Publication history

Received

Accepted

Published

Issue Date

DOI

https://doi.org/10.1038/s41567-017-0005-8

Further reading