# Noise-induced order

Noise-induced order is a mathematical phenomenon appearing in the Matsumoto-Tsuda[1] model of the Belosov-Zhabotinski reaction. The notable fact about this model is that 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:

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

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 ${\displaystyle T(0.3^{-})=T(0.3^{+})}$).

In [1] 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. It is worth remarking that the behavior of the floating point system and of the original system may differ [7], therefore [1] 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 [8]

## References

1. ^ a b c Matsumoto, K.; Tsuda, I. (1983). "Noise-induced order". J Stat Phys. 31 (1): 87–106. doi:10.1007/BF01010923.
2. ^ "Citation Details for "Noise-induced order"". Springer. doi:10.1007/BF01010923. 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. doi:10.1007/BF01041073.
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). 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: 014508. 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. 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.
8. ^ Galatolo, Stefano; Monge, Maurizio; Nisoli, Isaia (2017). "Existence of Noise Induced Order, a Computer Aided Proof". arXiv:1702.07024 [math.DS].