Lyapunov exponents for random maps
Lyapunov exponents for random maps
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DOI：
10.3934/dcdsb.2022058
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
F. Nakamura;Yushi Nakano;Hisayoshi Toyokawa
中科院分区：
文献类型：
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作者：
F. Nakamura;Yushi Nakano;Hisayoshi Toyokawa
最近人们认识到，对于紧流形上的丰富动力系统，李雅普诺夫指数不存在的点集（称为李雅普诺夫不规则集）具有正的勒贝格测度。在本文中，我们证明在任何物理噪声下，李雅普诺夫不规则集的勒贝格测度为零，并且此类李雅普诺夫指数的数量是有限的。该结果是关于时间平均值的存在性和有限性的阿劳霍定理的李亚普诺夫指数版本。此外，我们数值计算了具有吸引异宿连接的表面流的李亚普诺夫指数，该表面流在物理噪声下具有正勒贝格测度的李亚普诺夫不规则集。本文还证明了在非物理噪声下具有吸引同宿/异宿连接的表面流的正勒贝格测度集上李雅普诺夫不规则行为的消失。
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.