Noise-induced order

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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 background

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:

[math]\displaystyle{ T(x)=\begin{cases} (a+(x-\frac{1}{8})^{\frac{1}{3}})e^{-x}+b, & 0\leq x\leq 0.3 \\ c(10xe^{\frac{-10x}{3}})^{19}+b & 0.3\leq x\leq 1 \end{cases} }[/math]

where

  • [math]\displaystyle{ a=\frac{19}{42}\cdot\bigg(\frac{7}{5}\bigg)^{1/3} }[/math] (defined so that [math]\displaystyle{ T'(0.3^-)=0 }[/math]),
  • [math]\displaystyle{ b=0.02328852830307032054478158044023918735669943648088852646123182739831022528_{158}^{213} }[/math], such that [math]\displaystyle{ T^5 (0.3) }[/math] lands on a repelling fixed point (in some way this is analogous to a Misiurewicz point)
  • [math]\displaystyle{ c=\frac{20}{3^{20}\cdot 7}\cdot\bigg(\frac{7}{5}\bigg)^{1/3}\cdot e^{187/10} }[/math] (defined so that [math]\displaystyle{ T(0.3^-)=T(0.3^+) }[/math]).

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 also

References

  1. ↑ 1.0 1.1 Matsumoto, K.; Tsuda, I. (1983). "Noise-induced order". J Stat Phys 31 (1): 87–106. doi:10.1007/BF01010923. Bibcode: 1983JSP....31...87M. 
  2. ↑ Citation Details for "Noise-induced order". Springer. doi:10.1007/BF01010923. 
  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. doi:10.1007/BF01041073. Bibcode: 1989JSP....55..941D. 
  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. doi:10.1103/PhysRevE.67.066220. PMID 16241339. Bibcode: 2003PhRvE..67f6220Z. 
  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. doi:10.1063/1.2946710. PMID 18624484. Bibcode: 2008JChPh.129a4508Y. 
  6. ↑ Hudson, J.L.; Mankin, J.C. (1981). "Chaos in the Belousov–Zhabotinskii reaction". J. Chem. Phys. 74 (11): 6171–6177. doi:10.1063/1.441007. Bibcode: 1981JChPh..74.6171H. 
  7. ↑ Guihéneuf, P. (2018). "Physical measures of discretizations of generic diffeomorphisms". Erg. Theo. And Dyn. Sys. 38 (4): 1422–1458. doi:10.1017/etds.2016.70. 
  8. ↑ Galatolo, Stefano; Monge, Maurizio; Nisoli, Isaia (2020). "Existence of noise induced order, a computer aided proof". Nonlinearity 33 (9): 4237–4276. doi:10.1088/1361-6544/ab86cd. Bibcode: 2020Nonli..33.4237G. 
  9. ↑ Nisoli, Isaia (2020). "How does noise induce order?". arXiv:2003.08422 [math.DS].



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