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Oscillatory cluster patterns in a homogeneous chemical system with global feedback

Nature volume 406, pages 389–391 (2000)Cite this article

Abstract

Oscillatory clusters are sets of domains in which nearly all elements in a given domain oscillate with the same amplitude and phase1,2,3,4. They play an important role in understanding coupled neuron systems5,6,7,8. In the simplest case, a system consists of two clusters that oscillate in antiphase and can each occupy multiple fixed spatial domains. Examples of cluster behaviour in extended chemical systems are rare, but have been shown to resemble standing waves9,10,11,12,13, except that they lack a characteristic wavelength. Here we report the observation of so-called ‘localized clusters’—periodic antiphase oscillations in one part of the medium, while the remainder appears uniform—in the Belousov–Zhabotinsky reaction–diffusion system with photochemical global feedback. We also observe standing clusters with fixed spatial domains that oscillate periodically in time and occupy the entire medium, and irregular clusters with no periodicity in either space or time, with standing clusters transforming into irregular clusters and then into localized clusters as the strength of the global negative feedback is gradually increased. By incorporating the effects of global feedback into a model of the reaction, we are able to simulate successfully the experimental data.

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Figure 1: Experimental set-up.
Figure 2: Standing clusters.
Figure 3: Irregular and localized clusters.
Figure 4: Oscillatory cluster patterns in our BZ reaction–diffusion model with global negative feedback.

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References

  1. Golomb, D., Hansel, D., Shraiman, B. & Sompolinsky, H. Clustering in globally coupled phase oscillators. Phys. Rev. A 45, 3516–3530 (1992).

    Article  ADS  CAS  Google Scholar 

  2. Hakim, V. & Rappel, W.-J. Dynamics of the globally coupled complex Ginzburg-Landau equation. Phys. Rev. A 46, 7347–7350 (1992).

    Article  ADS  Google Scholar 

  3. Falcke, M., Engel, H. & Neufeld, M. Cluster formation, standing waves, and stripe patterns in oscillatory active media with local and global coupling. Phys. Rev. E 52, 763–771 ( 1995).

    Article  ADS  CAS  Google Scholar 

  4. Wang, W., Kiss, I. Z. & Hudson, J. L. Experiments on arrays of globally coupled chaotic electrochemical oscillators: Synchronization and clustering. Chaos 10, 248–256 ( 2000).

    Article  ADS  CAS  Google Scholar 

  5. Golomb, D. & Rinzel, J. Clustering in globally coupled inhibitory neurons. Physica D 72, 259– 282 (1994).

    Article  ADS  Google Scholar 

  6. Terman, D. & Wang, D. Global competition and local cooperation in a network of neural oscillators. Physica D 81, 148–176 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  7. Huerta, R., Bazhenov, M. & Rabinovich, M. I. Clusters of synchronization and bistability in lattices of chaotic neurons. Europhys. Lett. 43, 719–724 (1998).

    Article  ADS  CAS  Google Scholar 

  8. Rabinovich, M. I., Varona, P. & Torres, J. J. Slow dynamics and regularization phenomena in ensembles of chaotic neurons. Physica A 263, 405– 414 (1999).

    Article  ADS  Google Scholar 

  9. Lev, O., Sheintuch, M., Pismen, L. M. & Yarnitzky, Ch. Standing and propagating wave oscillations in the anodic dissolution of nickel. Nature 336, 458–459 (1988).

    Article  ADS  CAS  Google Scholar 

  10. Cordonier, G. A., Schuth, F. & Schmidt, L. D. Oscillations in methylamine decomposition on Pt, Rh, and Ir: Experiments and models. J. Chem. Phys. 91, 5374–5386 (1989).

    Article  ADS  CAS  Google Scholar 

  11. Jakubith, S., Rotermund, H. H., Engel, W., von Oertzen, A. & Ertl, G. Spatiotemporal concentration patterns in a surface reaction: Propagating and standing waves, rotating spirals, and turbulence. Phys. Rev. Lett. 65, 3013– 3016 (1990).

    Article  ADS  CAS  Google Scholar 

  12. Somani, M., Liauw, M. A. & Luss, D. Evolution and impact of temperature patterns during hydrogen oxidation on a Ni ring. Chem. Eng. Sci. 52 , 2331–2341 (1997).

    Article  CAS  Google Scholar 

  13. Petrov, V., Ouyang, Q. & Swinney, H. L. Resonant pattern formation in a chemical system. Nature 388, 655–657 (1997).

    Article  ADS  CAS  Google Scholar 

  14. Belousov, B. P. in Collection of Short Papers on Radiation Medicine 145– 152 (Medgiz, Moscow, 1959).

    Google Scholar 

  15. Zhabotinsky, A. M. Periodic liquid phase reactions. Proc. Acad. Sci. USSR 157, 392–395 (1964).

    Google Scholar 

  16. Zaikin, A. N. & Zhabotinsky, A. M. Concentration wave propagation in a two-dimensional, liquid phase self-oscillating system. Nature 225, 535–537 ( 1970).

    Article  ADS  CAS  Google Scholar 

  17. Winfree, A. T. Spiral waves of chemical activity. Science 175, 634–636 (1972).

    Article  ADS  CAS  Google Scholar 

  18. Ross, J., Müller, S. C. & Vidal, C. Chemical waves. Science 240, 460–465 (1988).

    Article  ADS  CAS  Google Scholar 

  19. Kapral, R. & Showalter, K. (eds) Chemical Waves and Patterns (Kluwer, Dordrecht, 1995).

    Book  Google Scholar 

  20. Kuhnert, L., Agladze, K. I. & Krinsky, V. I. Image processing using light-sensitive chemical waves. Nature 337, 244–247 (1989).

    Article  ADS  CAS  Google Scholar 

  21. Grill, S., Zykov, V. S. & Müller, S. C. Feedback-controlled dynamics of meandering spiral waves. Phys. Rev. Lett. 75, 3368– 3371 (1995).

    Article  ADS  CAS  Google Scholar 

  22. Kadar, S., Amemiya, T. & Showalter, K. Reaction mechanism for light sensitivity of the Ru(bpy)2+3-catalyzed Belousov-Zhabotinsky reaction. J. Phys. Chem. A 101, 8200–8206 (1997).

    Article  CAS  Google Scholar 

  23. Yamaguchi, T., Kuhnert, L., Nagy-Ungvarai, Zs., Müller, S. C. & Hess, B. Gel systems for the Belousov-Zhabotinskii reaction. J. Chem. Phys. 95, 5831–5837 (1991).

    Article  CAS  Google Scholar 

  24. Amemiya, T., Kettunen, P., Kadar, S., Yamaguchi, T. & Showalter, K. Formation and evolution of scroll waves in photosensitive excitable media. Chaos 8, 872– 878 (1998).

    Article  ADS  CAS  Google Scholar 

  25. 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).

    Article  CAS  Google Scholar 

  26. Bugrim, A., Dolnik, M., Zhabotinsky, A. M. & Epstein, I. R. Heterogeneous sources of target patterns in reaction-diffusion systems. J. Phys. Chem. 100, 19017–19022 (1996).

    Article  CAS  Google Scholar 

  27. Gao, Y. & Försterling, H.-D. Oscillations in the bromomalonic acid/bromate system catalyzed by [Ru(bipy)3]2+. J. Phys. Chem. 99, 8638– 8644 (1995).

    Article  CAS  Google Scholar 

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Acknowledgements

This work was supported by the Chemistry Division of the National Science Foundation and the W. M. Keck Foundation.

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Authors and Affiliations

  1. Department of Chemistry and Volen Center for Complex Systems, MS 015, Brandeis University, Waltham, 02454-9110, Massachusetts, USA

    Vladimir K. Vanag, Lingfa Yang, Milos Dolnik, Anatol M. Zhabotinsky & Irving R. Epstein

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  1. Vladimir K. Vanag
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Correspondence to Anatol M. Zhabotinsky.

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Vanag, V., Yang, L., Dolnik, M. et al. Oscillatory cluster patterns in a homogeneous chemical system with global feedback. Nature 406, 389–391 (2000). https://doi.org/10.1038/35019038

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