Lyapunov exponents for random maps

Fumihiko Nakamura; Yushi Nakano; Hisayoshi Toyokawa

doi:10.3934/dcdsb.2022058

Article Contents

2022, Volume 27, Issue 12: 7657-7669. Doi: 10.3934/dcdsb.2022058

This issue Previous Article Smoothing effect and well-posedness for 2D Boussinesq equations in critical Sobolev space Next Article The isothermal limit for the compressible Euler equations with damping

Lyapunov exponents for random maps

1.
Faculty of Engineering, Kitami Institute of Technology, Hokkaido, 090-8507, Japan
2.
Department of Mathematics, Tokai University, Kanagawa 259-1292, Japan

^*Corresponding author: Hisayoshi Toyokawa
^*Corresponding author: Hisayoshi Toyokawa

Received: April 2021

Revised: February 2022

Early access: March 2022

Published: December 2022

This work is partially supported by JSPS KAKENHI Grant Numbers 19K14575, 19K21834 and 21K20330.

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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.

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Mathematics Subject Classification: Primary: 37H15; Secondary: 37A25, 37H05.

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Figure 1. The finite time flow Lyapunov exponent for the system (7) are illustrated without noise (left), with additive noise (right). They are plotted by $ \log $ time scale

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Figure 2. The image of the model with perturbation of impulsive type

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