Lyapunov exponents for random maps

TL;DR

This paper investigates how physical noise affects the measure and existence of Lyapunov exponents in dynamical systems, showing that noise can eliminate irregular behavior and confirming theoretical results with numerical examples.

Contribution

It proves that physical noise reduces the Lyapunov irregular set to zero measure and finiteness of exponents, extending Araújo's theorem to stochastic settings.

Findings

Lyapunov irregular set has zero Lebesgue measure under physical noise

Number of Lyapunov exponents is finite with noise

Numerical computation confirms theoretical predictions

Abstract

It has been recently realized that for abundant dynamical systems on a compact manifold, the set of points for which Lyapunov exponents fail to exist, called the Lyapunov irregular set, has positive Lebesgue measure. In the present paper, we show that under any physical noise, the Lyapunov irregular set has zero Lebesgue measure and the number of such Lyapunov exponents is finite. This result is a Lyapunov exponent version of Ara\'{u}jo's theorem on the existence and finitude of time averages. Furthermore, we numerically compute the Lyapunov exponents for a surface flow with an attracting heteroclinic connection, which enjoys the Lyapunov irregular set of positive Lebesgue measure, under a physical noise. This paper also contains the proof of the disappearance of Lyapunov irregular behavior on a positive Lebesgue measure set for a surface flow with an attracting homoclinic/heteroclinic connection under a non-physical noise.