The first calculation performed was the Hurst Exponent
for the Saudi Basic Industries return from 09/21/2005 to 09/21/2010.
, as an important exponent
to test the property of chaotic map, is widely used in the world of chaos.
(2004) modeled the daily returns series of the PSI-20 using a formulation of fBm and determined the Hurst exponent
by DFA from 1993 to 2001.
The values of scaling exponent
in temperature anomaly records are significantly higher than those in precipitation.
From Figures 3 and 4, we can see that the Lyapunov exponents
of Figure 3 are less than 0 or tend to be 0, while in Figure 4, there is one exponent
that is greater than 200 with no tendency to 0.
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The Hurst exponent
can be understood as a scaling relation for statistically self-similar data .
Depending on the value of the Lyapunov exponent
there are three options:
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The top Lyapunov exponent
of the nonlinear system is evaluated by Wolf's algorithm to determine whether the chaotic phenomenon of the nonlinear system actually occurs.
A number of algorithms can be employed to estimate the power exponent
values, including the least squares method (LSM), maximum-likelihood method (MLM), and major axis method (MAM).
The theory of function spaces with variable exponent
was extensively studied by researchers since the work of Kovacik and Rakosnik  appeared in 1991; see [7,8] and the references therein.