Noise-induced order

Noise-induced order is a mathematical phenomenon appearing in the Matsumoto-Tsuda[1] model of the Belosov-Zhabotinski reaction.

In this model, adding noise to the system causes a transition from a "chaotic" behaviour to a more "ordered" behaviour; this article was a seminal paper in the area and generated a big number of citations[2] and gave birth to a line of research in applied mathematics and physics.[3][4] This phenomenon was later observed in the Belosov-Zhabotinsky reaction.[5]

Mathematical backgroundEdit

Interpolating experimental data from the Belosouv-Zabotinsky reaction,[6] Matsumoto and Tsuda introduced a one dimensional model, a random dynamical system with uniform additive noise, driven by the map:



  •   (defined so that  ),
  •  , such that   lands on a repelling fixed point (in some way this is analogous to a Misiurewicz point)
  •   (defined so that  ).

This random dynamical system is simulated with different noise amplitudes using floating-point arithmetic and the Lyapunov exponent along the simulated orbits is computed; the Lyapunov exponent of this simulated system was found to transition from positive to negative as the noise amplitude grows.[1]

The behavior of the floating point system and of the original system may differ;[7] therefore, this is not a rigorous mathematical proof of the phenomenon.

A computer assisted proof of noise-induced order for the Matsumoto-Tsuda map with the parameters above was given in 2017.[8] In 2020 a sufficient condition for noise-induced order was given for one dimensional maps:[9] the Lyapunov exponent for small noise sizes is positive, while the average of the logarithm of the derivative with respect to Lebesgue is negative.

See alsoEdit


  1. ^ a b Matsumoto, K.; Tsuda, I. (1983). "Noise-induced order". J Stat Phys. 31 (1): 87–106. Bibcode:1983JSP....31...87M. doi:10.1007/BF01010923. S2CID 189855973.
  2. ^ "Citation Details for "Noise-induced order"". Springer. doi:10.1007/BF01010923. S2CID 189855973. {{cite journal}}: Cite journal requires |journal= (help)
  3. ^ Doi, S. (1989). "A chaotic map with a flat segment can produce a noise-induced order". J Stat Phys. 55 (5–6): 941–964. Bibcode:1989JSP....55..941D. doi:10.1007/BF01041073. S2CID 122930351.
  4. ^ Zhou, C.S.; Khurts, J.; Allaria, E.; Boccalletti, S.; Meucci, R.; Arecchi, F.T. (2003). "Constructive effects of noise in homoclinic chaotic systems". Phys. Rev. E. 67 (6): 066220. Bibcode:2003PhRvE..67f6220Z. doi:10.1103/PhysRevE.67.066220. PMID 16241339.
  5. ^ Yoshimoto, Minoru; Shirahama, Hiroyuki; Kurosawa, Shigeru (2008). "Noise-induced order in the chaos of the Belousov–Zhabotinsky reaction". The Journal of Chemical Physics. 129 (1): 014508. Bibcode:2008JChPh.129a4508Y. doi:10.1063/1.2946710. PMID 18624484.
  6. ^ Hudson, J.L.; Mankin, J.C. (1981). "Chaos in the Belousov–Zhabotinskii reaction". J. Chem. Phys. 74 (11): 6171–6177. Bibcode:1981JChPh..74.6171H. doi:10.1063/1.441007.
  7. ^ Guihéneuf, P. (2018). "Physical measures of discretizations of generic diffeomorphisms". Erg. Theo. And Dyn. Sys. 38 (4): 1422–1458. arXiv:1510.00720. doi:10.1017/etds.2016.70. S2CID 54986954.
  8. ^ Galatolo, Stefano; Monge, Maurizio; Nisoli, Isaia (2020). "Existence of noise induced order, a computer aided proof". Nonlinearity. 33 (9): 4237–4276. arXiv:1702.07024. Bibcode:2020Nonli..33.4237G. doi:10.1088/1361-6544/ab86cd. S2CID 119141740.
  9. ^ Nisoli, Isaia (2020). "How does noise induce order?". arXiv:2003.08422 [math.DS].