Noisy hodgepodge machine and the observed mesoscopic behavior in the non-stirred Belousov–Zhabotinsky reaction

Optimal noise and hidden noise in the hodgepodge machine


In this paper, we have modified one of the simplest multi-level cellular automata – a hodgepodge machine, so as to represent the best match for the chemical trajectory observed in the Belousov–Zhabotinsky reaction (BZR) in a thin layered planar setting. By introducing a noise term into the model, we were able to transform the central regular structure into the circular target pattern. We further analyze influences of the neighborhood (diffusion process) and internal excitation type of noise. We find that the configurations of ignition points, which give the target patterns, occur only in the interval of the neighborhood excitation noise from 30% to 34% and at the internal excitation noise of 12%. We argue that the BZR occurs on a semi-regular grid – a chemical analogy to a Bénard cell in the viscous fluid, and we discuss the size of the relevant elementary cell. In this way, the BZR is a quintessential example of mesoscopic process, in particular, it does follow neither the deterministic rules of the microscopic system nor the tenet of Boltzmannian statistic physics that only the most frequent events are observed.

This is a preview of subscription content, log in to check access.


  1. 1.

    V.K. Vanag, Physics–Uspekhi 42, 413 (1999)

    ADS  Google Scholar 

  2. 2.

    M. Gerhardt, H. Schuster, J.J. Tyson, Science 30, 1563 (1990)

    ADS  Article  Google Scholar 

  3. 3.

    S. Hiltemann, Multi-coloured cellular automata, Bachelor thesis, Erasmus Universiteit, Rotterdam, The Netherlands, 2008

  4. 4.

    D. Štys, T. Náhlík, A. Zhyrova, R. Rychtáriková, Š. Papáček, P. Císař, in HPCSE 2015, Lect. Notes Comp. Sci. 9611, 171 (2016)

    Google Scholar 

  5. 5.

    A.K. Dewdney, Sci. Am. 225, 104 (1988)

    Article  Google Scholar 

  6. 6.

    J. García-Ojalvo, L. Schimansky-Geier, Europhys. Lett. 47, 298 (1999)

    ADS  Article  Google Scholar 

  7. 7.

    R. FitzHugh, Bull. Math. Biophys. 17, 257 (1955)

    Article  Google Scholar 

  8. 8.

    J. Nagumo, S. Arimoto, S. Yoshizawa, Proc. IRE 50, 2061 (1962)

    Article  Google Scholar 

  9. 9.

    A. Wuensche, Exploring discrete dynamics (Luniver Press, 2011)

  10. 10.

    A.V. Getling, Rayleigh-Bénard convection: structures and dynamics (World Scientific, 1998)

  11. 11.

    J. Cohen,

  12. 12.

    U. Wilensky, NetLogo B-Z Reaction model (Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL, 2003),

  13. 13.

    M. Blasone, P. Jizba, G. Vitiello, Quantum field theory and its macroscopic manifestations. Boson condensation, ordered patterns and topological defects (Imperial College Press, Cambridge, UK, 2011)

Download references

Author information



Corresponding author

Correspondence to Renata Rychtáriková.

Additional information

Supplementary material in the form of one zip file available from the Journal web page at

Electronic supplementary material

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Štys, D., Rychtáriková, R., Zhyrova, A. et al. Noisy hodgepodge machine and the observed mesoscopic behavior in the non-stirred Belousov–Zhabotinsky reaction. Eur. Phys. J. Spec. Top. 227, 2361–2374 (2019).

Download citation