1104
Prog. Theor. Phys. Vol. 71, No.5, May 1984, Progress Letters
Non-Stationarity of Chaotic Motions
in an Area Preserving Mapping
Tamotsu KOHY AMA
Department of Physics, Kyoto University, Kyoto 606
(Received December 26, 1983)
The power spectrum and the Allan variance are calculated numerically for the standard mapping. The
1-" spectrum is obtained in wide parameter regime. The sojourn time distribution around the main
resonance is clarified to be hyperbolic. These results show that the chaotic orbits are endowed with a kind
of non·stationarity.
II! spectrum and non-stationarity
The II! power spectrum is observed
experimentally in a wide variety of fluctuations as well
as the voltage fluctuation across a semiconductor
with direct electric flow.') It is also known that
the 1/I-type power spectrum appears in the critical regime of the intermittent chaos in a dissipative dynamical system. Z) On the other hand,
the properties of the power spectrum for a conservative dynamical system are not elucidated well.
The purpose of this paper is to study the statistical quantities like the power spectrum for the
stochastic region of the standard mapping and to
report the existence of the non-stationary behavior. We have calculated numerically the power
spectrum for the stochastic trajectories and observed the f-"-type spectrum for the wide parameter
regions. We have also calculated the Allan variance 3 ) and the sojourn time distribution which
characterize the chaotic trajectories in phase
space. The non-stationary behavior is observed
by these three statistical quantities as follows: the
power spectrum S(f)~ r v , and the Allan variance V (n) ~ n r, and the sojourn time distribution
P(m)~m-p. As is discussed in the previous
paper,') the termInology of "non-stationarity" is
defined by the following conditions: 1I~1, r~O, fJ
~ 2. We have observed that these conditions are
satisfied in the standard mapping for several
parameter regions. The hierarchical structure of
the island KAM tori in phase space is thought to
be responsible for these non-stationary characters.
Model system
The standard mapping describes the motion of
the pendulum under the parametric forcing by
periodic pulses, 5)
fi+l :Ii-~Sin(27rOi)'
0i+l-0i+li+l,
where 0 and I describe the phase and the angular
momentum of the pendulum respectively. The
mapping is the area preserving one with the
constant Jacobian. We adopted cos(271Bi) as a
physical variable of the system and calculated the
power spectrum and the Allan variance for the
time series {cos(271Bi) }r~l.
Power spectrum
We choose an initial point near the unstable
fixed point (0, 1)=(0.5, 0) and pursued the
chaotic behavior in the main stochastic region.
Some examples of the power spectrum S(f) are
shown in Fig. 1. S(f) were calculated by the
Fast Fourier Transformation of length 214 and
were averaged over 80 sample paths. Figure 1
indicates an example of the I-v -type power
spectrum (II"" 1) in the low frequency regime.
The saturation frequency Is of the I-v -type
spectrum is estimated to be smaller than 10- 5 for
these examples. The index II and the saturation
frequency Is are estimated for each parameter
K(Fig.2). II and Is oscillate as K increases and
both behaviors well accord· with each other.
Three minimums of II are observed in the region
1 < K < 2. The successive minimum value of II
seems to decrease as K increases. For instance,
when K is nearly equal to 1.1, II is about 1.1.
This implies that the time series is non-stationary
for a long time (t""'10 5 ). On the other hand,
when the parameter K is 1.9, the chaos is still
stationary since II is nearly 0.75. The anomalous
oscillatory dependence of II in Fig. 2 resembles
the oscillatory behavior of the diffusion coefficient
Progress Letters
May 1984
N
J
K=O. 6 [
~oj
~,.-='~:.'
I
j
~
1105
W\;
.,.
.•......
: -: '~"'-.~
o
I
..
',.. ;,.
I ...
'..
•
I:,
1
,""W"
" +- - - ,.-.~~~~~'''''io
'
-5
"
,
LD
I
I
..
..............
'.'
HNJ
I
.'
070------~-------2~------~3---
K
.. i
-4
-2
-3
-1
LOG{f)
"
K=1.4
o
Fig. 2.
-4
-3
, -2
o
-1
LOG{f)
Fig. 1.
Power spectrums for K
= 0.6 and
K
= 1.4.
D, which is observed in much larger parameter
region of K.6) For the parameter region where v
has a minimum value, the orbits in the phase
space stay for an extremely long time near the
boundary layers between the stochastic region
and the KAM tori around the fixed point or the
periodic points. The long time correlation seems
to be created from these sticky boundary layers.
The oscillatory dependence of v on K indicates
the periodic appearance of the sticky boundary
layers as K increases. When K is smaller than
unity, there appear many resonance island chains
of KAM tori in the phase space and many sticky
regions coexist.
Allan variance
The Allan variance for the same time series
{cos(271Bi) }r~l are calculated for which the power
spectrum is obtained. The Allan variance for the
time series {X ( i )} is defined as follows. 3 ) Setting
The index
l/
3
2
K
of I-V spectrum and saturation
frequency'ls as a function of K.
i=nk+j (O;;;;j;;;;n-l), we divide the series
{X(i)} into groups with n members and calculate
the mean value
1 n-l
Mn(k)=- L: X(nk+ j)
nj~O
for each group.
'1
The Allan variance is defined by .
1 N-l
V( n)=2»~ N ~o (Mn(k+ 1)- Mn(k»2 .
The dependence of the Allan variance V ( n ) ~ n'
(I': positive) characterizes the non-stationarity of
the time sequences. 4 ) The Allan variance V ( n )
is measured experimentally for. various fluctuations where the 1/f power spectrum is observed.
Especially, the flicker floor (1'=0) is often observed for the exact II! spectrum. 7 ) The n' type
Allan variance and the f- v type power spectrum
are strongly correlated with each other by
v=y+l.
In Fig. 3, we show the changes of the Allan
variances V (n) for the time series {cOS(21CBi) }r~l
around K = 1.5. The several characteristic time
Progress Letters
1106
Vol. 71, No.5
0
0
0
1. 50
K=O.6
"
OJ
",
.-a-.-..r._--.:t.....,.
0
-~~~;-.;:-
-
a.. '
u..
'", \.
o
\. 1.54
\
K=1.48 \
~~1------~2~------~3~------~4~----~~5
LOG (t)
Fig. 3.
o
'"
'",
Changes of Allan variance around K = 1.5.
domains in Fig. 3 are classified as follows:
I) quasi·periodic time domain: The trajectories in the phase space can be assumed to be
quasiperiodic, so that the Allan variance
V(n) is proportional to n- 2 •
II) weakly non-stationary time domian: This
domain contributes to the j-V type power
spectrum. The Allan variance V ( n) is almost
independent of n, i.e., y=d(In V(n))/d(In n)""O.
This flicker floor (y ~ 0) is not affected largely by
the appearance of the sticky region in the phase
space.
III) strongly non-stationary time domain: This
time domain appears when the stochastic region
in the phase space has the sticky boundary around
the resonance KAM island chains. Here, the
time series seems to be a strongly non-stationary
stochastic process with a relatively large positive
"o
~+---r--.---r--'---'~-'i---'~Ti--'---'
'0.00
0.80
1.60
2.40
3.20
q.oo
LOG OF N
o
'"
.
K=1.4
o
,
0,
<>
'D
'",
y.
IV) diffusion dominant time domain: For large
n, y becomes -1. This time domain can be well
described by the Gaussian process. The nonstationary time· domain (II and III) appears
beyond the limit of our calculation time (~10 5 ).8)
Sojourn time distribution
In order to investigate the statistical properties
of the sticky boundary of the stochastic region,
we have calculated numerically the distribution of
the number of successive rotations (sojourn time)
around the main resonant KAM torus. Figure 4
shows the hyperbolic distribution p( m)= m- Il
where m is the number of successive rotations
aroud the main KAM torus. The index /3 at K
=0.6 is nearly 1.8, which suggests the nonstationary behavior of the stochastic trajectories,
~+---r--.---r--,,---r--'---''--'i---r--'i
'0:00
0.80
1.60
2.'10
3.20
q.oo
LOG OF N
Fig. 4. Sojourn time distributions for K=O.6 and K
=1.4.
since the average number of rotations <m>
becomes very large. This result is consistent
with the non-stationary behavior obtained from
the power spectrum and the Allan variance. For
K = 1.4, the index /3 is larger than 2. In this case,
the motion around the main resonance recovers
the stationarity.
Discussion
We present the three indices jI, y, and /3 for K
=0.6. The index jI of the
type power
spectrum is jI = 1.24 ±.1. The Allan variance
rv
May 1984
Progress Letters
V( n) for 10 2 < n< 10 5 can be fitted to n' where r
=0.10±.05 The distribution P(m) of the number
of successive rotations around the main resonance
becomes hyperbolic P ( m) ~ m - P and the index fJ
is nearly equal to 1.8±.1. These three indices
suggest that the stochastic behavior of the stand·
ard mapping for K=0.6 has a kind of non·
stationary character. It is shown by Fig. 2 that
non·stationary states can be created in the appro·
priate parameter regions. The hyperbolic distri·
bution of the sojourn time around the main
resonance indicates that these long time behav·
iors can be attributed to the motions near the
boundary of the stochastic region. 9 ) Considering
the hierarchical structure of phase space, we
suspect that the infinitely many island chains of
KAM tori near the sticky resonance region are
responsible for these anomalous behaviors. In a
forthcoming paper, the statistical properties dis·
cussed in this paper will be elucidated theoretical·
ly taking account of the hierarchical structure of
the KAM tori in phase space.
1107
The author would like to thank Professor K.
Tomita for valuable discussions and Dr. Y. Aiz·
awa for stimulating conversations and critical
reading of the manuscript.
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(1983), 497.
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