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)