The addition of stochastic noise to a deterministic dynamical system can induce dramatic qualitative changes in its behaviour. In one counterintuitive effect – termed noise induced order – the addition of noise to an already chaotic system can make its behaviour more orderly and regular. This was first discovered in numerical experiments in a one-dimensional map model of the Belousov-Zhabotinsky reaction, where researchers found sharp changes in various dynamical measures – the Lyapunov exponent, Kolmogorov-Sinai entropy and the power spectrum – with increasing noise amplitude. Similar noise-induced order was later confirmed in physical Belousov-Zhabotinsky reactions and in other experiments. Other systems reveal more complicated noise-induced transitions, with dynamics moving back and forth between chaotic and regular dynamics with increasing noise.
In a recent paper, LML External Fellow Yuzuru Sato and colleagues examine multiple noise-induced transitions in Lasota-Mackey maps – a class of one-dimensional maps which originally emerged in the study of delay-differential equations arising in models of blood cell production. The authors draw on ideas recently exploited by other researchers who were able to prove – using validated numerics, or rigorous computation – the existence of noise-induced order in the Belousov-Zhabotinsky map. This work established a change in the sign of the Lyapunov exponent with increasing noise amplitude. Sato and colleagues use a similar approach to compute the Lyapunov exponents of the Lasota-Mackey map.
As they demonstrate, this map also displays a number of noise-induced transitions; indeed, rigorous computation demonstrates that the sign of Lyapunov exponent in the map changes at least three times. As a consistency check, the authors also compare the results of the rigorous computation with a non-rigorous computation, and confirm that the non-rigorous method well-approximates the Lyapunov exponents derived using the rigorous approach. As the computational algorithm employed here is known to be effective for a wide class of random dynamical systems with additive noise, the authors expect this approach should be useful in deriving strong conclusions about the statistical properties of a broad class of nonlinear stochastic phenomena.
The paper is available as a pre-print at https://arxiv.org/pdf/2102.11715.pdf