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 ^[1] and gave birth to a line of research in applied mathematics and physics. ^{[2] [3]} This phenomenon was later observed in the Belosov-Zhabotinsky reaction. ^[4]

Mathematical background

Interpolating experimental data from the Belosouv-Zabotinsky reaction, ^[5] Matsumoto and Tsuda introduced a one-dimensional model, a random dynamical system with uniform additive noise, driven by the map:

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

where

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

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; ^[6] 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. ^[7] In 2020 a sufficient condition for noise-induced order was given for one dimensional maps: ^[8] 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

• Self-organization
• Stochastic Resonance

References

1. 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. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. Nisoli, Isaia (2023). "How Does Noise Induce Order?". Journal of Statistical Physics. 190 (1): 22. arXiv: 2003.08422 . Bibcode:2023JSP...190...22N. doi:10.1007/s10955-022-03041-y. ISSN   0022-4715.