Energy and Power Engineering, 2013, 5, 46-50
doi:10.4236/epe.2013.54B009 Published Online July 2013 (
Analysis of Chaotic Characters for the Monthly Runoff
Series at Fudedian Station in Liaohe Bain
Haiying Hu1, Huamao Huang2
1School of Civil and Transportation Engineering, South China University of Technology, Guangzhou, P. R. China
2School of Science, South China University of Technology, Guangzhou, P. R. China
Received September, 2012
The evolution of monthly runoff is affected both by climate environment and human activities, and its characteristics
play an important role in runoff prediction and simulation. In this paper, the G-P and the principal component analysis
method, which are both based on the reconstruction theory of the phase space, are used to study the chaos characteris-
tics of the monthly runoff series at Fudedian station in Liaohe basin. The results show that the monthly runoff series
have a large probability of chaos.
Keywords: Chaos Analysis; Saturated Correlation Dimension; Principal Component Analysis; Monthly Runoff Series
1. Introduction
Over the past two decades, the theory of chaos has shown
its applicability to a wide class of scientific problems in
various fields, including astronomy, biology, chemistry,
ecology, engineering, and physics. The discovery of
chaos, starting with Lorenz [1], provides a helpful dis-
cussion. Casdagli [2] presents an effective comparative
study between chaotic and stochastic models. The dis-
covery that even very simple deterministic systems can
produce seemingly irregular outputs (e.g. time series)
provided an impetus to seek possible chaotic patterns in
real systems, through applications of a host of nonlinear
dynamic and chaos methods.
Chaotic studies are necessary for a complex or non-
linear system. Based on the chaotic theory, it is well
known that a random-like series can be attributed to de-
terministic rules. In fact, with the development of the
chaotic theory, chaotic study in geoscien ces has not only
been restricted to the basic but it has also come into the
practical study stage, such as the characteristics of cha-
otic evolution, dynamical system reconstruction and non-
linear prediction. The discovery and continuing research
of fractals and deterministic chaos gives hydrology re-
search a new theorem, methods and opportunities.
Therefore, the application of the concept of chaos theory
in hydrology has been gaining considerable interest in
recent times [3-5].
Runoff change is very complex, which is the combined
result of climate environment and human activities. This
paper investigates the existence of chaotic behavior in
the monthly runoff time series at the Fudedian station in
Liaohe basin, China. Monthly data observed over a pe-
riod of 39 years (1964-2004) are studied. Four nonlinear
dynamic methods, with varying levels of complexity, are
employed: (1) phase space reconstruction; (2) autocorre-
lation function; (3) correlation dimension meth od; and (4)
principal component analysis method. These methods
provide either direct or indirect identification of chaotic
2. Study Area
Liaohe River located in the southwest of northeast China,
is one of China's seven major rivers. The Liaohe River
Basin lies between 40°30'- 45°10'N and 117°00'-
125°30'E. The climate of the Liaohe Basin is chiefly
characterized by monsoonal weather conditions with the
annual average temperature of 5℃ to 11. The region
of this study lies between Fudedian and Tieling Station in
Liaohe basin, with an area of 13839 km2. Affected by the
atmospheric circulation, precipitation in the study region
is very unevenly distributed in time and space, which is
mainly concentrated in the summer season with 60-70%
of the annual precipitation. Rainstorm lasts shorter and
rainfall concentrates. Consequently, floods in the study
area are frequent and serious.
3. Methodologies
3.1. Phase Space Reconstruction
The first step in the process of chaotic analysis is that of
Copyright © 2013 SciRes. EPE
H. Y. HU, H. M. HUANG 47
attempting to reconstruct the dynamics in phase-space.
Monthly runoff series can be decided by many factors,
and a number of variables will be required when we use
equations to describe the process of its evolution. Gener-
ally, it is believed that the time series of a variable is th e
overall performance of the system, so the chaotic behav-
ior of the system can be studied by investigating the sin-
gle variable time series which can decide the long-term
evolution of the system.
The nature of dynamics of a real-world system may be
stochastic, deterministic or in between. This can be iden-
tified, at least as a preliminary indicator, by using the
phase space concept. A method for reconstructing a
phase-space from a single time series has been presented
by Takens [6]. The Takens theorem states that the un-
derlying dynamics can be fully recovered by building an
m-dimensional space. The dynamics of a scalar time se-
xx x are embedded in the
m-dimensional phase-space. The phase-space is defined
 
,,,1 1,2,Yixi xiximiN
  ,
1Nn m
 , τ is the time delay, m is the
embedding dimension, and
xx xx with
N-observed values.
To determine the fractal dimension of a dynamical
system, one should first decide the time delay and the
embedding dimension for the correlation dimension
analysis. Usually, the time delay is made with the help of
the autocorrelation function and the embedding dimen-
sion can be obtained by the G-P method [7]. Correlation
dimension can be used to estimate the sufficient embed-
ding dimension and the estimated fractal dimension. A
strange attractor could also be revealed in a chaotic sys-
tem under a phase space reconstruction environment.
For a monthly runoff time series 123
xx x
 , the
reconstruction of the phase-space is defined by:
111 1(1)
222 2(1)
lll lm
Yxx x
Yxx x
Yxx x
1ln m
 .
3.2. Autocorrelation Function
The purpose of using autocorrelation function in chaotic
analysis is to help select a proper time delay. The time
delay can’t be too small, or too large. The choice of a too
small time delay causes information overlap among sub-
sequent datum. On the other hand, the choice of a too
large time delay can lead to loss of all relevant informa-
tion in phase space reconstruction because neighboring
trajectories diverge. Therefore, the selection of a suitable
time delay would allow unfolding of the attractor in the
phase space while the components of any state vector
must be as uncorrelated as possible.
A good choice of the time delay is essential for geo-
metrical and numerical analysis of the phase space re-
construction. For the time series 123
xx x
 , the
Autocorrelation function is defined as follows:
where, τ is the time delay,
is the mean and
standard deviation of the time series, respectively. It may
be chosen as the lag time at which the auto-correction
becomes zero. However, considering various values of τ
demonstrates that the results do not show a strong de-
pendence on the actual value chosen.
3.3. G-P Method
There are few distinct methods for computing fractal
dimensions: relative dispersion analysis, correlation
analysis, Fourier analysis and rescaled range analysis. To
estimate the fractal dimension of a time series, the con-
cept of correction dimension is useful. Correlation di-
mension is a nonlinear measure of the correlation be-
tween pairs lying on the attractor. Correlation dimension
estimation is related to the relativ e frequency with which
the attractor visits each covering element. Correlation
dimension is generally a lower bound measure of the
fractal dimension.
The correlation dimension can be measured by the G-P
method suggested by Grassberger and Procaccia [7]. The
main steps of the G-P method are:
1) For a time series 123
xx x , selecting a proper
time delay τ and embedding dimension m and recon-
structing the phase space.
2) The correlation integral is then calculated,
which is given by
Crr Y Y
is the Heaviside step function, ij
urYY 
for , and , for 0u
. N is
the number of points on the reconstructed attractor, r is
the radius of the sphere centered on i or Y
Y. ij
is the Euclidean distance between the elements of
and i
3) For a large number of points, the correlation inte-
gral follows the power law:
Copyright © 2013 SciRes. EPE
Cr r
is a constant; and D is the correlation dimen-
sion, which represents generally good estimate of the
fractal dimension of the attractor. The slope of
versus is given by:
ln r
The behavior of provides one technique for de-
termining th e presence o f ch aos in a time series, such that
(i) for stochastic processes, , varies linearly with in-
creasing m, without reaching a saturation value; (ii) for
deterministic processes the value of saturates after a
certain value of m. The saturation value is the fractal
dimension of the attractor or the time series.
4) Gradually increasing the phase space dimension m
and repeating the first three steps, calcu lating the estima-
tion of systematic dimension D(m) until the D(m) is not
obviously changed while increasing the phase dimension
m. That means the D(m) has tended to the saturation.
3.4. Principal Component Analysis Method
The principal component analysis method is able to ef-
fectively identify the chaos and noise. In this method, the
covariance of the trajectory line matrix formed in
the reconstruction phase space is calculated first, and
then the eigenvalues and the corre-
sponding eigenvectors of the covariance matrix
are calculated later. Eigenvalue can be
arranged by size as follows:
1, 2,,
1, 2,,
Ui m 
 .
The eigenvalues i
and the eigenvectors are
principal components. The sum of all the eigenvalues
is obtained as:
The map with the axes of the index i and
ln i
referred to the primary component spectrum, and the
primary component spectrum of the chaotic sequence is a
straight line or containing a linear portion, with negative
slope and though the vertex. Thus the chaotic sequence
can be identified by use of this property.
4. Analysis and Results
In this paper, the monthly runoff time series at Fudedian
station in Liaohe basin, China is used and analyzed to
investigate the possible existence of chaotic behavior.
The average monthly runoff is 6.0 m3/s. Figure 1 shows
the variation of monthly runoff obtained from Fudedian
4.1. Determination of the Delay Time
The quality of a phase space portrait depends on the time
delay and therefore, a reasonable value of the time delay
is desired. According to the formula of autocorrelation
function method, monthly runoff time series at Fudedian
station are analyzed and the correlation function value
with the change of the delay time can be obtained as
shown in Figure 2. It is seen from Figure 2 that the cor-
relation function value of monthly runoff time series at
Fudedian station beco mes zero for the first time while the
delay time is 4. Therefore, the delay time in phase space
reconstr uc tion can b e de termin e d as τ = 4.
4.2. Calculation of the Saturation Correlation
Saturation correlation dimension is calculated b y the G-P
method. The embedding dimension m is set to 4, 5, 6, 7,
8, 9, 10, 11, 12, and 13. The r e su l t can be seen in Figures
3 and 4. Figure 3 shows the behavior of correlation
function lnC(r) against lnr for values of increasing m.
Figure 4 shows that the correlation dimension values
D(m) increase with increasing the embedding dimension
Figure 1. The variation of monthly runoff at Fudedian sta-
Figure 2. Autocorrelation function plot for monthly runoff
series of Fudedian Station.
Copyright © 2013 SciRes. EPE
H. Y. HU, H. M. HUANG 49
m, up to a certain value and then reaches a plateau, where
its value saturates. This is an indication of deterministic
dynamics and the saturated correlation dimension is 2.66
for monthly runoff data at Fudedian Station. The nearest
integer above the correlation dimension value (D = 3) is
taken as the minimum dimension of the phase space that
can embed the attractor. The value of m at the saturation
point (m = 10) is supposed to provide the sufficient
number of variables to describe the dynamics of the at-
4.3. Principal Component Analysis
The monthly runoff time series at Fudedian station are
analyzed by the principal component analysis method
and the relation of dimension i and PCA for monthly
runoff series of Fudedian station can be seen in Figure 5.
It is seen from Figure 5 that there is a straight line
with negative slope in the relation diagram of embedded
dimension i and PCA for monthly runoff series of Fud-
edian station. The result sh ows that monthly runoff ser ies
Figure 3. Relational curves of lnC(r) ~ lnr for monthly run-
off series of Fudedian Station.
Figure 4. Relational curves of D(m) ~ m for monthly runoff
series of Fudedian Station.
Figure 5. The relation of dimension i and PCA for monthly
runoff series of Fudedian station.
have chaotic characteristics. This further confirmed that
the monthly runoff series is a chaotic sequence.
5. Conclusions
This paper investigates possible chaotic behaviors in the
monthly runoff at the Fudedian station in Liaohe basin,
China. The autocorrelation function, the correlation di-
mension and the principal component analysis method
are used in the analysis. The results from these methods
provide convincing indication and confirmation of the
existence of a mild low-dimensional chaos in monthly
runoff ser ies for the data used. On the basis of the attrac-
tor dimension, the minimum number of variables essen-
tial to model the dynamics of the monthly runoff of the
Fudedian station in Liaohe basin is identified as three and
the number of variables sufficient is ten.
6. Acknowledgements
This paper is supported by the National Natural Sciences
Foundation of China (grant number: 51209096), the
Open Funding of the State Key Laboratory of Hydrol-
ogy-Water Resources and Hydraulic Engineering of Ho-
hai University (grant number: 2011490411), and the
Fundamental Research Funds for the Cen tral Universities,
SCUT(grant number: 2011ZZ0017).
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