Abstract
For an n-dimensional chaotic system, we extend the definition of the nonlinear local Lyapunov exponent (NLLE) from one- to n-dimensional spectra, and present a method for computing the NLLE spectrum. The method is tested on three chaotic systems with different complexity. The results indicate that the NLLE spectrum realistically characterizes the growth rates of initial error vectors along different directions from the linear to nonlinear phases of error growth. This represents an improvement over the traditional Lyapunov exponent spectrum, which only characterizes the error growth rates during the linear phase of error growth. In addition, because the NLLE spectrum can effectively separate the slowly and rapidly growing perturbations, it is shown to be more suitable for estimating the predictability of chaotic systems, as compared to the traditional Lyapunov exponent spectrum.
摘 要
非线性局部 Lyapunov 指数(NLLE)方法目前已经在天气和气候可预报性研究中得到了广泛应用. 不过, 由于非线性正交化处理的困难, 过去我们仅仅定义了一维的 NLLE, 其描述了误差最快发展方向上的非线性增长率. 对于低维混沌系统来说, 系统的误差增长率主要取决于误差在最快发展方向的增长率. 但是, 对于高维复杂的混沌系统可以同时容纳不同增长率的误差增长; 除了最快发展方向的误差增长, 其它发展方向的误差增长同样也可以导致系统出现大的预报误差. 因此, 为了完全展示大气等复杂非线性系统的动力学特征, 充分表征复杂系统中误差在不同发展方向上的非线性增长率, 我们将一维的NLLE拓广到多维的情形, 建立了 NLLE 谱, 发展了对应于多个误差增长方向上的非线性误差增长理论. 通过应用于三个不同复杂度的混沌模型, 结果表明: 对于 n 维非线性动力系统, NLLE 谱可以很好地表征 n 个误差增长最快到增长最慢(或者收缩最快)方向上的初始误差在有限时间内的非线性增长率. 通过NLLE谱随时间的演化, 表明不同误差增长方向上误差的增长率随时间并不是一成不变的, 在误差增长的线性阶段和非线性阶段有明显不同的特征, 比传统的全局 Lyapunov 指数谱仅反映了误差增长的线性阶段特征有明显的优越性. 利用NLLE谱可以更有效地定量度量不同误差增长方向上混沌系统可预报期限的大小, 克服了传统的 Lyapunov 理论在衡量混沌系统可预报期限的局限性, 为定量估计高维复杂混沌系统可预报期限开辟了新的途径.
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Acknowledgements
This research was jointly supported by the National Natural Science Foundation of China for Excellent Young Scholars (Grant No. 41522502), the National Program on Global Change and Air–Sea Interaction (Grant No. GASI-IPOVAI- 06), and the National Key Technology Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2015BAC03B07).
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Ding, R., Li, J. & Li, B. Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system. Adv. Atmos. Sci. 34, 1027–1034 (2017). https://doi.org/10.1007/s00376-017-7011-8
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DOI: https://doi.org/10.1007/s00376-017-7011-8