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

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.

2. ^ "Citation Details for "Noise-induced order"". Springer. doi:10.1007/BF01010923. S2CID 189855973. {{cite journal}}: Cite journal requires |journal= (help)