Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system

Advances in Atmospheric Sciences, Aug 2017

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.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2Fs00376-017-7011-8.pdf

Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system

Determining the Spectrum of the Nonlinear Local Lyapunov Exponents in a Multidimensional Chaotic System Ruiqiang DING 1 2 3 4 Jianping LI 0 1 3 Baosheng LI 1 3 4 5 0 College of Global Change and Earth System Sciences, Beijing Normal University , Beijing 100875 , China 1 Chengdu University of Information Technology , Chengdu 610225 , China 2 Plateau Atmosphere and Environment Key Laboratory of Sichuan Province 3 Institute of Atmospheric Physics, Chinese Academy of Sciences , Beijing 100029 , China 4 State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics 5 University of Chinese Academy of Sciences , Beijing 100049 , China 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. Lyapunov exponent; nonlinear local Lyapunov exponent; predictability 1. Introduction The estimation of the average rates of divergence (or convergence) of initially nearby trajectories in phase space has been studied with (global) Lyapunov exponents, which are used to quantify the average predictability properties of a chaotic system (Eckmann and Ruelle, 1985; Wolf et al., 1985) . The sum of all the positive Lyapunov exponents is an estimate of the Kolmogorov entropy K, and the inverse of K is a measure of the total predictability of the system (Kolmogorov, 1941; Fraedrich, 1987, 1988) . Considering that the (global) Lyapunov exponents only provide a measure of the total predictability of a system, various local or finite-time Lyapunov exponents (Nese, 1989; Houtekamer, 1991; Yoden and Nomura, 1993; Ziehmann et al., 2000) have been subsequently proposed to measure the local predictability around a point x0 in phase space. However, the existing global or local Lyapunov exponents have limitations because they all satisfy the assumption that the initial perturbations are sufficiently small and the tangent linear model (TLM) of a nonlinear system could approximately govern their evolution (Lacarra and Talagrand, 1988; Feng and Dong, 2003; Mu and Duan, 2003; Duan and Mu, 2009) . If an initial perturbation is large enough to invalidate the TLM, it is no longer possible to apply the existing global or local Lyapunov exponents in predictability studies of chaotic systems (Kalnay and Toth, 1995) . In view of the limitations of the existing global or local Lyapunov exponents, Ding and Li (2007) introduced the concept of the nonlinear local Lyapunov exponent (NLLE). The NLLE measures the average growth rate of the initial errors of nonlinear dynamical models without linearizing the governing equations. The experimental results of Ding and Li (2007) show that, compared with a linear local or finitetime Lyapunov exponent, the NLLE is more appropriate for the quantitative determination of the predictability limit of a chaotic system. Based on observational or reanalysis data, the NLLE method has been used to investigate the atmospheric predictability at various timescales (Ding et al., 2008, 2010, 2011, 2016; Li and Ding, 2008, 2011, 2013) . However, recall that the NLLE defined by Ding and Li (2007) only characterizes the nonlinear growth rate of the initial perturbations along the fastest growing direction, which is insufficient to describe the expanding or contracting nature of the initial perturbations along different directions in phase space. This explains why the Lyapunov exponent spectrum, rather than only the largest Lyapunov exponent, was introduced in the traditional Lyapunov theory. Therefore, for an n-dimensional chaotic system, it is necessary to extend the definition of the NLLE from one- to n-dimensional spectra, as this would allow us to investigate nonlinear evolution behaviors of initial perturbations along different directions in phase space. In this paper, we first introduce the definition of the NLLE spectrum, and then propose a method to compute the NLLE spectrum. Finally, we demonstrate the validation and usefulness of the NLLE spectrum in characterizing the nonlinear evolutionary behaviors of initial perturbations along different directions, and in measuring the predictability limit of chaotic systems, by applying it to three chaotic syst (...truncated)


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00376-017-7011-8.pdf

Ruiqiang Ding, Jianping Li, Baosheng Li. Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system, Advances in Atmospheric Sciences, 2017, pp. 1027-1034, Volume 34, Issue 9, DOI: 10.1007/s00376-017-7011-8