Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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268
Spatiotemporal Chaos in Distributed Systems:
Theory and Practice
George
P.
Pavlos, A. C. Iliopoulos,
V.
G.
Tsoutsouras,
L.
P. Karakatsanis, and
E.
G.
Pavlos
Department
of
Electrical and Computer Engineering
Democritus University
of
Thrace, Xanthi, Greece
(e-mail:
gpavlos@ee.duth.gr)
Abstract: This paper presents theoretical and experimental results concerning the
hypothesis
of
spatiotemporal chaos in distributed physical systems far from equilibrium.
Modern tools
of
nonlinear time series analysis, such as the correlation dimension and the
maximum Lyapunov exponent, were applied to various time series, corresponding to
different physical systems such as space plasmas (solar flares, magnetic-electric field
components) lithosphere-faults system (earthquakes) brain and cardiac dynamics during
or
without epileptic episodes. Futhermore, the method
of
surrogate data was used for the
exclusion
of
'pseudo chaos' caused by the nonlinear distortion
of
a purely stochastic
process. The results
of
the nonlinear analysis presented in this study constitute
experimental evidence for significant phenomena indicated by the theory
of
nonequilibrium dynamics such as nonequilibrium phase transItIOn, chaotic
synchronization, chaotic intermittency, directed percolation, defect turbulence, spinodal
nucleation and clustering.
Keywords:
nonlinear time series analysis, spatiotemporal chaos, Self Organized
Criticality
(SOC), distributed systems, nonequilibrium dynamics.
1.
Introduction
In
this study we are interested in distributed nonlinear physical systems
such as space plasmas, earthquakes, brain and cardiac dynamics which
include practically infinite degrees
of
freedom,
as
well
as
the
theoretical presuppositions for understanding their complex dynamics.
At the meso scopic level, these systems are described by variables
which are functions
of
space and time as coarse grained fields
of
microscopic variables. Also, the spontaneous formation
of
the
spatiotemporal structures is the result
of
sudden qualitative changes
known as bifurcations, which occur
if
an externally controllable driving
passes through critical values. At bifurcation points the system can
perceive its environment and adapt to it by forming preferred patterns
of
modes
of
behaviour. According
to
Wilson, 1979 [15], precisely at
the critical point, the scale
of
the largest fluctuations becomes infinite,
but the smallest fluctuations are in no way diminished. The theory that
Spatiotemporal Chaos
ill
Distributed Systems 269
describes a physical system near its critical point must take in account
the entire spectrum
of
length scale.
Recently, for the understanding
of
such distributed physical
systems two theoretical concepts, contradicting for many scientists,
have been introduced: the self organized criticality
(SOC) and the low
dimensional chaos. In a series
of
papers we have supported
experimentally and theoretically the broad universality
of
SOC and low
dimensional chaotic processes for distributed physical systems. In this
study, we present new experimental results and theoretical concepts
that reveal the deeper and universal character
of
the nonequilibrium
phase transition and the critical point theory characteristics, applied to
distributed physical systems. Particularly, in section 2 we introduce
significant theoretical concepts concerning the far from equilibrium
phase transition process. In section
3,
we present new results obtained
by modem nonlinear time series analysis applied to data concerning
five different physical systems, such
as
the solar system, the earth's
magnetosphere and crust, the human brain and cardiac dynamics. In
section 4 we summarize crucial results obtained by the data analysis
and in section 5, we give theoretical explanations
of
our results
embedded in the modem theory
of
spatiotemporal chaos.
2. Far from Equilibrium Physical Processes
The far from equilibrium nonlinear dynamics includes significant
collective phenomena such as: power law distribution and critical scale
invariance, nonequilibrium fluctuations causing spontaneous nucleation
and evolution
of
turbulent motion from metastable states, defect
mediated turbulence and localized defects changing chaotically in time
and moving randomly in space, spatiotemporal intermittency, chaotic
synchronization, directed percolation causing levy-flight spreading
processes, turbulent patches and percolation structures, threshold
dynamics and avalanches, chaotic itinerancy, stochastic motion
of
vortex like objects.
In particular, in the phase space
of
a distributed system
various finite - dimensional attractors can exist such as fixed points,
limit cycles or more complicated finite dimensional strange attractors,
which correspond to chaotic dynamics as well
as
states known as self
organized critical states (SOC). Generally, spatiotemporal chaos
includes early turbulence with low effective dimensionality and few
coherent spatial patterns or states
of
fully developed turbulence. These
states are out
of
equilibrium steady states related to the bifurcation
270
G.
P.
Pavios et
ai.
points
of
the nonlinear distributed dynamics,
as
well as to out
of
equilibrium second order phase transition processes. Also, spatial and
temporal patterns as well as spatially localized structures are possible
solutions
of
the nonlinear spatially extended dynamics. Solar flares,
magnetospheric super-substroms, earthquakes and epileptic seizures
could be some paradigms
of
physical systems which include dynamical
coherent localized structures underlying these physical events.
2.1. The modern unified theory
of
the
far
from equilibrium physical
processes
A distributed system is governed by a generalized Langevin stochastic
equation
a;;
=.t:
(<p,x,t)+n;
(x,t)
(1)
Typical examples
of
such an equation are: the generalized Ginrburg-
Landau equation, the Maxwell-Boltzman plasma equations, the
Navier-Stokes equations, or the type
of
Kardak - Parisi - Zhang
stochastic partial differential equations. The continuous stochastic
processes
of
Eq. (1) can also be the field theoretical approach
of
coarse-
graining for (contact) processes such as cellular automata (CA) models
including threshold systems and avalanche type processes or coupled
Map Latticies(CML). In this coarse-grained approach the threshold
type coupling corresponds to continuous diffusion process.
The probabilistic description
of
the stochastic Langevin equation (1) is
given by the solution
of
the corresponding master equation or Fokker-
Plank equation for which the solution is
as
follows
p(<p(x,t))=
fD(x)exp{-i.
fL(<i>,<p,x)dx}dt
(2)
obtained after path integral intergration. In Eq. (2), P
(<p
(x, t))
corresponds to the propability density functional and L (
<i>
,
<p,
x) to the
stochastic Langrangian
of
the system. The application
of
the far from
equilibrium renormalization group theory (R.G.) at the stochastic
Langrangian reveals fixed (critical) points as well
as
relevant and
Spatiotemporai Chaos
ill
Distributed Systems
271
macroscopically dynamical parameters (order parameters)
of
the
system, that drives its dynamics near the fixed point. Also, scaling
(power laws) near criticality are caused by the combined effects and
long correlations
of
the fluctuations (Chang, 1992[3]).
The dynamic renormalization group shows that nonlinear
stochastic systems can exist near critical states and can exhibit low
dimensional chaotic or high dimensional self organized criticality
(SOC) and fractal behaviour with intermittent turbulent character. The
intermittent turbulence may grow producing more and larger coherent
structures as well
as
fluctuations and new resonance sites, where
random dynamic fields are extremely long ranged and there exist many
correlation scales.
According to these theoretical concepts, the solar flares, the
magnetospheric superstorm-substorm events, the earthquakes in earth's
crust, the epileptic seizures in the human brain or the cardiac
arrhythmia can be related to the stochastic behaviour
of
nonlinear
distributed dynamical processes.
3. Results
In this section, we present significant results obtained by the nonlinear
time series analysis apllied to signals concerning different physical
systems such as solar flares, magnetospheric plasma, earthquakes,
epileptic human brain and cardio rhythm. For an extended description
of
the algorithms
of
nonlinear time series analysis (geometrical and
dynamical characteristics) see Pavlos
et
al. 2003[9],2007[10].
3.1. Solar flares
Figure l(a) represents the daily Flare Index
of
the Solar activity that
was determined using the final grouped solar flares obtained by
NGDC.
It
is calculated for each flare using the formula: Q =
(i
*
t)
,
where "i" is the importance coefficient
of
the flare and "t" is the
duration
of
the flare in minutes. To obtain final daily values, the daily
sums
of
the index for the total surface are divided by the total tome
of
observation
of
that day. The data covers time period from 1/1/1996 to
31112/2007.
272
G.
P Pavlos et
al.
160
150
~~.,-----,--.--,------,--...,--,------,
120
~
80
c:
40
3-
16
0.4
4000
-0.5 0
0.5
1000
8000
Time
(Days)
m=IO,6=500
~
Solar Flares
--~
Surrogates
1.5 2 2.5
Ln (r)
2000 3000
Time
(Days)
12000 16000
(b)
3.5 4 4.5
4000 5000
(u)
100
50
·50
100
·150
+--,-----,--.--,----....,.-...,-~-__j
4 -
3 -
2 -
(f)
I
1000
4000
0000
Time
(Days)
m=7.6=20
--SUITog
ltc
12000
----.-
First
Difference
Solar
Flare
I
2
Ln(r)
I I
16000
(c)
l,..,i
:
;:;.
-" I -
I I
2000
3000
Time
(Days)
I
4000
-
-
5000
Figure
1.
a) Time Series
of
Solar Flares (Daily). b) Slopes
of
the correlation integrals
for
the Solar Flares Time series and its 30 surrogates.
c)
The largest Lyapunov exponent
(Lmax)
of
the Solar Flares time series as a function
of
time and their 30 surrogates.
d)
Time series
of
the First Difference Solar Flares (Daily). e) Slopes
of
the correlation
integrals estimated
for
the First Difference Solar Flares time series and its 30
Surrogates. f) The largest Lyapunov exponent (Lmax)
of
the First Difference Solar Flares
time series as a function
of
time and their 30 surrogates.
Figure 1 (b) illustrates the slopes
of
the correlation integral
estimated for the
Solar Flares time series and its 30 surrogates using
delay time r= 500 and embedding dimension m=lO,
as
a function
of
Ln(r). As it can be seen from this figure, there is no significant
difference between the original series and the surrogate data.
Similar
results are obtained estimating the Maximum Lyapunov exponent
Spatiotemporal Chaos
in
Distributed Systems
273
(Lmax), shown in Fig. l(c), for the Solar Flares time series and its 30
surrogates using delay time r= 500 and embedding dimension
m=
1 0, as
a function
of
time (days). These results support the stochastic character
of
the original signal and indicate the possibility
of
correlated noise.
Fig.
led) represents the First Difference filtered Solar Flare time series.
We used the high pass first difference filter in order
to
exclude the low
frequency component. Figures l(e-f) show the slopes
of
the correlation
integral and the Lmax using delay time r=20 and embedding dimension
m=7, for the First Difference Solar Flares time series and its 30
surrogates, correspondingly. As it can be seen in Fig. lee) there is a low
value saturation
D
of
the slopes D
""
2 for a sufficient large embedding
dimension
m=7, revealing low dimensional profile
of
the underlying
dynamics. Moreover, Fig. l(f) shows the existence
of
a positive Lmax,
indicating sensitivity to initial conditions for the underlying system. In
Figs. 1 (e-
f)
we also show the clear and significant difference between
the First Difference Solar Flares time series and the surrogate data. The
above results support the hypothesis
of
the existence
of
a strange
attractor underlying the Solar flare dynamics, revealed after using the
first difference filter.
3.2. Magnetosphere
In this section we present results for three time series concerning the
Earth's Magnetosphere plasma sheet obtained by the spacecraft
GEOT AIL. The data period corresponds to a Superstorm which took
place at 27 July
of
2004. In particular, we estimate geometrical
(correlation dimension) and dynamical (maximum Lyapunov exponent)
characteristics in the reconstructed state space
of
the Magnetic field (Bz
- component), Electric field (Ex - Component), Ion moments (Velocity
- Vy).
Figure 2(a) illustrates the Magnetic Field First Difference
Filtered (FDF) Bz-component time series. In Fig. 2(b) the slopes
of
the
correlation integral are presented, estimated for the (FDF) Bz time
series and its surrogates using delay time
r=1O,
embedding dimension
m=9, and Theiler's parameter w=lO, as a function
of
Ln(r). Fig. 2(c)
shows the Lmax using delay time
r=
10
and embedding dimension
m=9, as a function
of
time (days). Fig. 2(d) represents the Electric field
(Ex - Component). The slopes
of
the correlation integral and the
maximum Lyapunov exponent
of
the Ex time series and its surrogates,
are shown in Figs. 2(3-f). The parameters we used were delay time
r=ll,
w=17 and embedding dimension m=9. Figures 2(g-i) represent
274
G.
P.
Pavlos et al.
the time series
of
Ion Moment Velocity (Vy-component), the slopes
of
the Correlation Integral estimated for the Vy-component series and its
Surrogate using
m=8, r=13, w=13 and the Lmax
of
the Vy-Component
series and its Surrogates using
m=8, r=13. The results indicate the
existence of chaotic dynamics corresponding to (FDF) Bz, Ex, Vy time
series, as well
as
the existence
of
high dimensional dynamics
corresponding
to
Bz time series, underlying the Magnetosphere.
~
Firsl
DiHerooce
FiHer
Time
Series
,.,
"
r
-t1lMt"IIM~."~1-I
~
.
! · '0
+-~
-,
--~
-,
~~
o _ _
Time
(3
sec
reso
l
ution)
(04:C1O
UT
·
17:00
UT
27
10712(0
4)
Ln(r)
Ti
me
o <(I(JQ
""""
,
~
Time(3se<:resolu~oo)
(04:00· 17:00UT. 24
107120(4)
o
1000
2IW
J(IOO
Timep2
secresoiution)
((14
:
0010
1
7:00
,
24J{)712004)
~.
,
lo(l)
Time
Time
Figure
2.
a) First Difference Filtered (FDF) Time Series
of
Magnetic Field (Bz-
Component). b) Slopes
of
the Correlation Integral estimated
for
the FDF Bz-Component
series and its Surrogate.
c)
Lmax
of
the FDF Bz-Component series and its Surrogate
s.
d)
Time Series
of
Electric Field (Ex-Component).
e)
Slopes
of
the Correlation Integral
estimated
for
the Ex-Component series and its Surrogate. f) Lmax
of
the Ex-Component
series and its Surrogates. g) Time Series
of
Ion Moment Velocity (Vy-Component). h)
Slopes
of
the Correlation Integral estimated
for
the Vy-Component Series and its
Surrogate.
i) Lmax
of
the Vy-Component series and its Surrogates.
3.3. Earthquakes
In this section results obtained by applying nonlinear analysis to three
seismic data sets are presented. The data series correspond to
5400
seismic events that have occurred in the Hellenic region, using a
threshold local magnitude
of
4 Richter and were observed during the
time period
of
1968-2004, within the broad Hellenic area 34.00-42.00
Spatiotemporal Chaos in Distributed Systems 275
Nand
18.00-30.00
E.
These data series, which correspond
to
the basic
focal parameters (time, space, magnitude)
of
earthquakes, were
constructed by the bulletins
of
the National Observatory
of
Athens. In
Figure 3(a,d,g) the three time series
of
seismic events are illustrated,
which correspond
to
the interevent series (time intervals between
successive earthquakes (Fig.
3a», the Latitude (Fig. 3d) and the
Magnitude (Fig. 3g). In these figures, it can be observed that the signals
maintain a strong stationary and aperiodic profile.
Figures 3(b-e) present the slopes
of
the correlation integral
estimated for the Interevent and the Latitude time series and their
corresponding surrogate data, using delay time
r=l,
and embedding
dimension
m=6
as
a function
of
Ln(r), correspondingly.
As
it can be
seen these slopes reveal efficient scaling and low saturation value,
indicating few degrees
of
dynamical freedom whereas there
is
a
significant discrimination between the two first Oliginal signals and
their surrogates. However, in the case
of
Magnitude series, shown in
Fig. 3(h), the slopes estimated using delay time
"j
m:::(i,(F1
5
----e-
lot
ero
'en! TIIIlI:
Series
1 :
.j
~~
___________
!!!!!!!I.-;;
.S
:!i;;;
__
,
'-j
0-1
+-:
-,--,,-r-,-r,
-''--'-'
.
~-,--,---l
l.5
3.5
l.n(r)
Ln(r)
m=6.60-1
--.-
Utiludc&,;.,.
-S'mGt~k-'
lex
2-.A1
n.'!l
.000
~~
.;oc.J
Scil-11lio:E."t\I1Ii
Figure
3.
a) Interevent seismic time series. b) Slopes
of
the
co
rrelation integral estimated
for interevent time series and its
30 surrogates.
c)
Lmax
of
the Interevent time series and
its
30 surrogates.
d)
Latitude time series. e) Slopes
of
the correlation integral estimated
for Latitude time series and
its30 surrogates.
j)
Lmax
of
the Latitude time series and its
30 surrogates.
g)
Magnitude time Serie
s.
h) Slopes
of
the Correlation Integral estimated
for the Depth Time
Series. i) Lmax
of
the Depth Time Series and its 30 Surrogates.
276
G.
P.
Pavlos et
al.
r=1
and embedding dimensions
m==3-8
as
a function
of
Ln(r), do not
reveal efficient scaling and saturation indicating strong stochastic
dynamics. Figs. 3(
c,
f,i)
illustrate the Lmax
of
the three time series
using parameters
r=1,
m=6,
7, correspondingly.
As
it is shown, the
Lmax is positive for all cases but therc exists a significant difference
between the original time series and their surrogates, only
fo
r the
Interevent and Latitude time series and not for the Magnitude series.
The results indicate the coexistence
of
low chaotic and high
dimensional dynamics underlying the Hellenic region.
3.3.
HW1lan
brain and cardiac rhythm
In
general, epileptic seizures may
an~hythmias.
The present study
Electrocncephalography (EEG)
be associated with cardiac
was aimed at evaluating
Figure
4.
a)
EEG
time series
of
patient with
Ollt
epileptic seizure. b) Stopes
of
the
Correlation Integral estimated
for
the
EEG
series and its Surrogate (thick line). c)
ECG
time series of' patient without epileptic seizure. d) Slopes
of
the Correlation Integral
estirrwtedfor the
ECG
series and its Surrogate (thick line).
Spatiotemporal Chaos
in
Distributed Systems 277
(b)
~t"
l
I
10'
Figure
5.
a) EEG time se
ri
es
of
patient during epileptic seiz
ure.
h) Slopes
of
the
Correlation Integral estimated for the EEG
series and its Surrogate (thick line). c) ECG
time series
of
patient during epileptic seizure. d) Slopes
of
the Correlation Integral
estimated for the
ECG series and its Surrogate (thick line).
and electrocardiographic (EeG) changes occurring before and during
partial epileptic seizures. We describe the features
of
the brain and
cardiac patterns that may have implications for understanding cardiac
and neuroautonomic instability in epilepsy.
Figure 4(a) presents an
EEG (brain dynamics) time series of a
patient without epileptic seizure. The duration
of
EEG segment is 20
sec (4000 points). Figure 4(b) represents the slopes
of
the correlation
integral estimated for the
EEG series using parameters m=6-1O, r=22,
w=22 and its surrogate using parameters
m=lO, r=22,
w=
22 (thick
line). Figures
(c
-
d)
are similar to figs. 4(a-b) but correspond
to
an
EeG
(electrocardiographic) time series
of
a patient without epileptic seizure.
The duration
of
the
EeG
segment is 20 sec (4000 points) and the slopes
of
the correlation integral were estimated using
m=6-1O
, r=6, w=6 for
the
EeG
time series and m=lO, r=6, w=6 for its Surrogate (thick line).
In
particular, Figs. 5 (a-d) are similar to figs. 4(a-d) but
represent an
EEG and an
EeG
time series
of
a patient during epileptic
seizure. The parameters used for the estimation
of
the slopes were
m=6-1O
, r=7, w=7 for EEG time series and m=lO, r=7, w=7 for its