Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

TL;DR

This paper demonstrates that small random perturbations of predominantly expanding multimodal circle maps lead to positive Lyapunov exponents and unique stationary measures, with verifiable finite-time criteria depending logarithmically on the perturbation size.

Contribution

It provides a finite-time, checkable criterion for positive Lyapunov exponents in randomly perturbed multimodal maps, contrasting with uncheckable conditions in deterministic chaos.

Findings

Positive Lyapunov exponents under random perturbations

Existence of unique stationary measures for perturbed maps

Finite-time criterion depending on logarithm of perturbation size

Abstract

We study the effects of IID random perturbations of amplitude $\epsilon > 0$ on the asymptotic dynamics of one-parameter families $\{f_a : S^1 \to S^1, a \in [0,1]\}$ of smooth multimodal maps which "predominantly expanding", i.e., $|f'_a| \gg 1$ away from small neighborhoods of the critical set $\{ f'_a = 0 \}$. We obtain, for any $\epsilon > 0$, a \emph{checkable, finite-time} criterion on the parameter $a$ for random perturbations of the map $f_a$ to exhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponent comparable to $\int_{S^1} \log |f_a'| \, dx$. This stands in contrast with the situation for the deterministic dynamics of $f_a$, the chaotic regimes of which are determined by typically uncheckable, infinite-time conditions. Moreover, our finite-time criterion depends on only $k \sim \log (\epsilon^{-1})$ iterates of the deterministic dynamics of $f_a$, which grows quite slowly as $\epsilon \to 0$.