Lyapunov exponents for random maps
arXiv:2103.11531 · doi:10.3934/dcdsb.2022058
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 Araújo'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.
arXiv comment14 pages, 2 figure
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