Noise-induced order
Template:Short description 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:
where
- (defined so that ),
- , such that lands on a repelling fixed point (in some way this is analogous to a Misiurewicz point)
- (defined so that ).
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.