Noise-induced order

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.

• Self-organization
• Stochastic Resonance