Suneet Dwivedi, A.K.Mittal, A.C.Pandey

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• Suneet Dwivedi, A.K.Mittal, A.C.Pandey
• M.N.Saha Centre of Space Studies,
• Institute of Interdisciplinary Studies,
• University of Allahabad , Allahabad 211002

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PART I

• A
• Paradigmatic Model
• for
• Study of
• Climate Change
• due to
• Forcing

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Main Points and Outline

• Paradigmatic model for discussing long-range
  monsoon predictability.
• The role of anomalies like Pacific sea surface
  temperature in Monsoon Variability.
• The local and global bifurcation structure in the
  r-F plane of the forced Lorenz Model.
• Shift in the probability distribution function
  between the two branches of the Lorenz attractor
  as a function of the forcing.

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Paradigmatic Lorenz Model

• Palmer (Palmer, 1994) introduced constant
  forcing terms in the classical Lorenz Model
  (Lorenz, 1963) to put forward a paradigmatic
  model for discussing long-range monsoon
  predictability.
• The forcing terms in this model correspond
  to the tropical Pacific sea surface
  temperature anomaly.

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Paradigmatic Lorenz Model(contd. )

•   ?The two branches of the Lorenz attractor
  correspond to the regimes of active and weak
  spells of the monsoon.
• ? In the absence of forcing, both the
  branches are equally likely corresponding to
  average rainfall. When forcing is introduced,
  the probability of the state lying in one of
  the branches is greater than in the other branch
  leading to excess / deficit rainfall.

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• Mathematical analysis
• Forced Lorenz system
• dx/dt -ax ay cFx
• dy/dt -xz rx - y cFy
  ( 1 )
• dz/dt xy - bz cFz
•  
• dx/dt -ax ay Fx
• dy/dt -xz rx - y Fy
  ( 2 )
• dz/dt xy - bz
•  

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• ? The fixed points of (2) are obtained easily in
  two cases
• Case (i) Fx aF, Fy -F and Case (ii)
  Fx aF, Fy -rF.
• For the case( i ) , the fixed points of
  (2) are O(0,-F,0) and P? (x?, y?, r -1),
• where,
• x? F ? ?F2 4b(r-1)/2
  
  (3 a)
• and y? x - F
  
  (3 b)
•  
• For r gt1, O is an unstable fixed point. If
  a gt (b 1), in the unforced case
• F 0, as r is increased through rc
  , the equilibrium points P? lose
• their stability via a Hopf bifurcation.
  However, stability can be restored at P
• to F gt Fc , and at P- to F lt -Fc where,
• 
  
  (4)
• Here. Fc approaches -? as r approaches rs
  from above. We have plotted Fc as a function of
  r, taking the standard values a 10 and b 8/3.

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P (P-) is locally stable if F gt Fc ( F lt -Fc )
Figure 1
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•   ?    We see that for r gt rc,
• the equilibrium point P (P-), which was
  unstable in the absence of forcing, can be made
  stable by a sufficiently large positive
  (negative) forcing parameter.
• For rs lt r lt rc,
• P (P-), which is stable in the absence of
  forcing, can be made unstable by a
  negative (positive) forcing parameter of
  sufficiently large magnitude. For 1
  lt r lt rs , the equilibrium points P? are stable
  for all values of the forcing
  parameter F.
• ?   For the case (ii), the fixed points of (2)
  are O (F,0,0) and P? (x?, y?, z?),
• where,
• (x? , y? ,z? ) ??b(r-1) , ??b(r-1)-
  F , (r-1)-(? F?(r-1)/b) (5)
• The equilibrium point O becomes
  stable if F gt Fc or F lt -Fc, where
• 
  (6)

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? We show a plot of Fc as a function of r for
r gt 1, taking the standard values a
10 and b 8/3. In the figure, rI is the
point of intersection of all three lines
of eqn. (10), its value is rI 5.
Fc
-Fc
O is locally stable if F gt Fc or F lt -Fc
Figure 2
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• ? We see that in contrast with case (i)
  where the fixed point O remains
• unstable for r gt 1, for all values of the
  forcing parameter, in case (ii) the
• fixed point O becomes stable for
  sufficiently large values of the forcing
• parameter magnitude .The equilibrium point
  P (P-) is stable if F lt Fc
• ( F gt -Fc ) where,
• 
  
• (7)
• ? Taking the standard values of a and b, we have
  plotted Fc as
• a function of r, for r gt 1. We see that
  for r gt rc, the
• equilibrium point P (P-), which was
  unstable in the
• absence of forcing, can be made stable by
  a sufficiently
• large negative (positive) forcing
  parameter. For 1 lt r lt rc,
• P (P-), which is stable in the absence of
  forcing, can be
• made unstable by a positive (negative)
  forcing parameter of
• sufficiently large magnitude.

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• Simulation results
• A unique feature of the Lorenz
  model is the
• co- existence of a strange attractor
  and two stable
• fixed points when the parameter r
  has values in the
• interval 24.06 lt r lt 24.74. We know
  that for r 28
• and F 0 , there is only one
  strange attractor and
• all the fixed points are
  unstable . At F Fc ,
• the fixed point P becomes stable.
• ? It is naturally of interest to know
  whether there is a
• parameter range of F for which there is
  a coexistence
• of a strange attractor and a stable
  fixed point.
•  

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P (P-) is locally stable if F lt Fc ( F gt
-Fc).Figure 3 It is interesting to note that
the value of r1 in fig3 is the same as for O in
fig 2.
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Figure 4
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• The regions shown in the figure are
  distinguished by the following properties
•  
• S.A There exists a strange attractor. None of
  the fixed points is stable. (For large
• values of r, stable periodic orbits are
  expected to exist, but this has not been
• investigated)
• I There exists a strange attractor. The
  fixed point P is locally stable. The
• unstable manifold of O belongs to the
  basin of attraction of the strange
• attractor.
• II There exists a Strange Attractor. The
  fixed point P- is locally stable. The
• unstable manifold of O belongs to the
  basin of attraction of the strange
• attractor.
• III There exists a Strange Attractor. Both
  the fixed points P and P- are stable.
• The unstable manifold of O belongs to the
  basin of attraction of the strange
• attractor.
• IV The fixed point P is stable. The unstable
  manifold of O belongs
• to the basin of attraction of P.
• V The fixed point P- is stable. The unstable
  manifold of O belongs to
• the basin of attraction of P-.

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• VI The fixed points P and P- are locally
  stable. An orbit starting at a point O- (O)
  which is slightly left (right) of O on the
  unstable manifold of O, will converge to P (P-).
  
• VII The fixed points P and P- are locally
  stable. The unstable manifold of O belongs to the
  basin of attraction of P.
• VIII The fixed points P and P- are locally
  stable. The unstable manifold of O belongs to the
  basin of attraction of P-.
• IX The fixed point P and P- are locally
  stable. An orbit starting at a point O- (O)
  which is slightly left (right) of O on the
  unstable manifold of O, will converge to P- (P).

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Zoomed portion of the box region in figure 5
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IX

Figure 5

Full bifurcation diagram for case (ii)
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? The probability of finding a point in the x gt 0
half-space i.e. p for the case (i) is plotted as
a function of forcing parameter F.
Figure 6
Probability Density
Function for case (i)
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Figure 7Probability Density Function for case
(ii)
? Here we determined the probability for
finding a point in x gt F halfspace and this has
been plotted as a function of forcing parameter.
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•  ? We studied the effect of forcing on the
  Lorenz
• map in an attempt to understand / predict
  the
• shift in PDF.
•  
•  ? Another interesting thing, which we
  obtained, is
• that the single cusp obtained in the
  Lorenz map
• in the absence of forcing splits into two
  cusps
• on introduction of forcing.

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Lorenz map of Lorenz attractor (Fx aF, Fy -
F)

• Figure 8 ( i )

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Lorenz map of Lorenz attractor (Fx aF, Fy
-rF)

• Figure 8 ( ii )

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• Conclusions
• ? We have done the mathematical analysis and a
  detailed study of the Forced Lorenz model and we
  find that the analysis is particularly simple if
  the forcing is varied along two particular lines
  in the FxFy plane and interesting if r is also
  varied along with forcing.
• ? For the case (i) the point (0, Fy, 0) is an
  unstable fixed point and for the case (ii) the
  point (-Fy , 0 , 0) becomes unstable fixed point
  only in the case when conditions (6) do not hold
  good.
•  ?  The local and global bifurcation structure
  has been studied extensively in the r-F plane for
  both the cases.

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• ? It is well known that the two symmetric fixed
  points of the unforced Lorenz model lose
  stability via a sub-critical Hopf bifurcation at
  a critical value rc. We find that for each r gt
  rc, there exists a critical value Fc of the
  forcing parameter, such that one of these points
  becomes locally stable if F gt Fc and the other
  becomes locally stable if F lt -Fc.
•  
• ? We find that the probability distribution
  function ( PDF ) shifts, from one branch of the
  Lorenz attractor to the other branch, linearly
  for small values of the forcing parameter F. Near
  a critical value of the forcing the probability
  of one of the branches suddenly approaches unity.
  
•  
• ? The Lorenz map, that is the maxima in z
  one-dimensional return map, which is known (
  Sparrow , 1982) to be a cusp in the absence of
  forcing, splits into two cusps for small values
  of the forcing.

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• ? Over and above, the coupled GCMs (AOGCM)
  failed miserably whereas AGCMs and OGCMs can
  give better forecasts. Climate drift is seen to
  occur when an AGCM and an OGCM are coupled into a
  single interactive system.
• ? The method developed here may be used as
  additional verification tool for coupled models
  as we have studied the effect of the forcing on
  probability distribution function and likely
  transition from one regime to another.
• ? Mathematical analysis and simulation results
  presented here will provide greater insight into
  the Forced Lorenz Model, which is of relevance to
  Monsoon Predictability.

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PART II

• Is there a Strange Attractor for Antarctic
  Oscillation?

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Main points and Outline

• A comprehensive study on atmospheric
  oscillations namely NAO, NPO, SO and AO is
  carried out to for understanding the local to
  planetary scale climate anomalies.
•   Chaotic time series analysis of time series
  data of all four oscillations indexes is done to
  look for the possible strange attractor for them.
•   Method of finding the dimensionality of single
  variable time series is used to find the possible
  cause behind the observed aperiodic behaviour.

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•   No strange attractor exists for NAO and AO
  whereas we can get strange attractors with
  fractal dimensionalities for SO and PDO / NPO.
• There is a need of development of improved
  techniques for estimating important geophysical
  parameters from satellite and in-situ data
  observations, and of more number of data stations
  in the Antarctic region.

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Introduction
Analysis of large-scale variabilities of
climate system of the atmosphere, ocean and sea
ice in the Indian and Pacific sectors around the
Antarctica focusing on data analyses is a
challenging area. Inter-annual and
decadal-scale fluctuations account for a large
percentage of the sea-ice and atmospheric
variability in the Arctic and Antarctica, as
revealed by the available observational records.
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The analysis of several year records of monthly
sea-ice concentration, sea-ice drift and sea
level pressure data around Antarctica reveals the
dominance of coupled ice-atmosphere oscillations
on the quasi-quadrennial timescale (periods of
3-5 yr), particularly significant in the Weddell
Sea and in the Ross-Amundsen-Bellingshausen Seas.
This is an expression of coupled air-ice-sea
interactions. Such oscillations are
significantly correlated with the El Nino
Southern Oscillation (ENSO) phenomenon.
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Pressure anomalies originating in the western
tropical Pacific on ENSO timescales are suggested
to reach the sub polar region through atmospheric
tele-connections. Thus, the natural climate
variability in both the Arctic and the Antarctic
regions is shown to involve regional coupled
interactions between the ocean, the sea-ice and
the atmosphere, which are strongly linked to
global-scale circulation patterns such as the NAO
in the Arctic and ENSO in Antarctica.
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Recent studies of the variability of surface
air temperature and sea-ice climatologies in the
western Antarctic Peninsula Region shows
long-term persistence in surface-air temperature
and sea-ice anomalies, where two to four
low-temperature/high-ice years are followed by
one to three high-temperature/low-ice years, a
pattern coherent with the Southern Oscillation
Index (SOI), indicating that there may be an ENSO
teleconnection between the western Antarctic
Peninsula region and lower latitudes.
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In order to look for the possible
teleconnection between different oscillations
found in the southern and northern hemisphere, we
have done the chaotic time series analysis of the
different oscillation indexes. The objective
of time series analysis is to build a model for
the unknown dynamical system that generated the
time series informative of Chaotic System because
often the only information we have about these
types of systems is in the form of single
variables time series.
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Different types of Oscillations

• 1) Southern Oscillation ( SO )
• 2) Pacific Decadal Oscillation (PDO)/ North
  Pacific Oscillation (NPO)
• 3) North Atlantic Oscillation (NAO)
• 4) Antarctica Oscillation (AO)

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Different types of Oscillations 1) Southern
Oscillation ( SO )
? The Southern Oscillation Index (SOI) is based
on the mean sea level pressure difference between
climate stations Tahiti, French Polynesia (149W,
14S) and Darwin, Australia (130.8E, 12.4S)
i.e. SOI (Tahiti - Darwin). When there is a
positive number, we have a La-Niña (or ocean
cooling), but when the number is negative we have
an El-Niño (or ocean warming). ? When the SO is
coupled with warming of the ocean off Peru and
Ecuador (El Nino) the resulting El Nino/Southern
Oscillation (ENSO) event can effect weather and
precipitation over much of the Tropics and
Subtropics.
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? The El Niño Southern Oscillation phenomenon (
ENSO) is the major cause of year-to-year
variations in climate over lower latitudes and
one of the most significant causes of global
climate change on this timescale.   ? This occurs
in a quasi-regular cycle, which has a variable
period of three to seven years meaning thereby
that this phenomenon is less chaotic and this
understanding is gradually leading to a
predictive capability.
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2) Pacific Decadal Oscillation (PDO) / North
Pacific Oscillation (NPO)
? "Pacific Decadal Oscillation" (PDO) is a
long-lived El Niño-like pattern of Pacific
climate variability. While the two climate
oscillations have similar spatial climate
fingerprints, they have very different behavior
in time. Two main characteristics distinguish PDO
from El Niño/Southern Oscillation (ENSO) i) 20th
century PDO "events" persisted for 20-to-30
years, while typical ENSO events persisted for 6
to 18 months. ii) The climatic fingerprints
of the PDO are most visible in the North
Pacific/North American sector, while secondary
signatures exist in the tropics - the opposite is
true for ENSO.
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? PDO index is derived as the leading PC of
monthly SST anomalies in the North Pacific Ocean,
poleward of 200 N. ? Causes for the PDO are not
currently known. Likewise, the potential
predictability for this climate oscillation is
not known. ? From a societal impacts
perspective, recognition of PDO is important
because it shows that "normal" climate conditions
can vary over time periods comparable to the
length of a human's lifetime.
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3) North Atlantic Oscillation (NAO)
? The North Atlantic Oscillation (NAO) / Arctic
Oscillation is a seesaw pattern in which
atmospheric pressure at polar and mid latitudes
fluctuates between positive and negative phases.
? The negative phase brings higher than normal
pressure over the polar region and lower than
normal pressure at about 45 degrees north
latitudes. The positive phase brings the opposite
conditions, steering ocean storms farther north
and bringing wetter weather to Alaska, Scotland
and Scandinavia and drier conditions to areas
such as California, Spain and the Middle East.
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? In recent years NAO has been mostly in its
positive phase and there is some speculation that
this may be a sign of human induced global
warming.
4) Antarctica Oscillation (AO)
? Antarctic oscillation (AO) refers to a
large-scale alternation of atmospheric mass
between the mid-latitudes and high latitudes
surface pressure.
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? In order to understand the variability of AO
and its possible connection with southern climate
anomalies in detail, an objective index of the AO
is defined which is known as Antarctic
Oscillation Index (AOI) .
? AOI is defined as the difference of zonal mean
sea level pressure between 400S and 650S. AOI has
the potential for clarifying climate regimes in
the southern hemisphere, similar to how the NAO
and NPO have been used in the Northern
Hemisphere.
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Correlation Dimension

• ? The dimension of an attractor is the first
  level of knowledge necessary to characterize its
  properties.
• ? The dimension is also a lower bound on the
  number of essential variable needed to describe
  (model) the dynamics of the system.
• ? Chaotic attractors have a highly fractured
  character unlike the simple attractors, which
  have regular structures. Here the dimension takes
  on values that are typically not integers.

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• Grassberger and Procaccia Algorithm
• Grassberger and Procaccia suggested a different
  measure for the strangeness of the attractors, a
  measure that can be easily obtained from any time
  series. The measure is obtained by considering
  correlation between points of a long time series
  on the attractor i.e. it is obtained by the
  correlation between random points on attractor.

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• Due to the exponential divergence of the
  trajectories, most pairs ( Xi , Xj ) with
• of a time series will be
  dynamically uncorrelated pairs of essentially
  random points. The points lie however on the
  attractor. Therefore they will be spatially
  correlated. This spatial correlation is measured
  with the Correlation Integral C(r), defined as
• 
  .. ( 1 )
• where is Heavyside function .

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?The central aim of the Grassberger and Procaccia
was to establish that for the small rs , C(r)
behaves like a power of r
.. ( 2 ) This Correlation exponent d
can be taken as a most useful measure of the
local structure of strange attractor. ? Given an
experimental signal, if one finds equation (2)
with d lt m , where m is embedding dimension , one
knows that the signal stems from deterministic
chaos rather than random noise, since random
noise will always result in C( l ) l m. With a
random noise, the slope of log C (r) vs. log r
will increase indefinitely as m is increased. For
a signal that comes from a strange attractor
(Chaotic signal), the slope will reach a value of
d saturates and will then become m independent.

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Optimised box assisted algorithm
? Analysing time sequence of length becomes
non trivial because a straightforward
computation of Correlation dimension of an
attractor from a time sequence of length N needs
a time of order . This time is further
enhanced if the estimate is to be made not only
for a single embedding dimension, but also for an
entire range. Thus a practical limit is reached
somewhere near N . ? An optimised
box assisted algorithm can be faster than the
algorithm given by Grassberger and Procaccia.
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? Here in addition to the big array, which needs
exactly N elements of N data points, we need an
array called BOX of the size of the mesh,
which provides the information where the
individual lists start in Llist (say). More
precisely, each element of BOX contains a pointer
to the head of the list for the box. It is empty
if no such list exists yet. Method for filling of
the boxes At the beginning all the elements of
BOX and Llist are set to zero. Then the data
points are read in, one after the other. Assume
that the kth data point falls into the box (I, J)
.If BOX (I, J) is still zero at this time, then
we set BOX (I, J) k, and read in the next data
point. If however the BOX (I, J) kp 0 then we
go to the kpth element of the Llist. If this is
zero, then we set Llist ( kp ) k , otherwise ,
if Llist ( kp ) is kp10 , we go to Llist (
kp1 ). Thus we follow the list until we reach the
first empty place, set this equal to k, and read
in the next data point.
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In order to find all near neighbours of the kth
data point, we go in two nested for and end loops
through the neighbouring boxes of BOX (I, J) ,
including this box itself. Whenever corresponding
element of box is kp not equal to zero, we first
of all know that the kpth data point is a
candidate for a near neighbour. In addition, if
Llist ( kp ) is kp1 not equal to 0 , then x( kp1
) is also a candidate , and so on . Thus by
walking through the linked list of the
neighbouring boxes, we collect all candidates for
near neighbours. The loop over the embedding
dimension mEmbed is the innermost and is
implemented via a conditional statement.
       
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Example 1 In a mesh of boxes, the first
data point has fallen into box (2,
4), points and has fallen into box
(4, 4), into box (4, 6) and into box (6, 6).
All empty elements are zero. Since box (4, 4) is
already filled with an entry 2 and hence
Llist (2) 4.
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4
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
Llist x
3 5

1 2



BOX
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Example 2 We implement our program on randomly
chosen points -
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Results and discussion
? The time series, which we took for our
analysis, are i) NAO Index, which is,
normalized pressure difference between a station
on the Azores and one on Iceland. This time
series is monthly anomaly for the period
1821-2000. ii) SO Index, which is, normalized
pressure difference between Tahiti and Darwin.
This time series is monthly anomaly for the
period 1866-2002.
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iii) NPO Index, which is derived as the leading
PC of monthly SST anomalies in the North Pacific
Ocean, poleward of 200N.This time series is
monthly anomaly for the period 1900-2001.
iv) AO Index, which is characterized by pressure
anomalies of one sign centered in the Antarctic
and anomalies of the opposite sign centered about
40-50S. AO Index is the dominant pattern of
non-seasonal tropospheric circulation variations
south of 30S. This time series is monthly anomaly
for the period 1948-2001.
 
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? In order to find the answer for this
aperiodicity, we have calculated the Correlation
dimension of all the four time series indexes
using the above-discussed algorithms.
? A plot between log C (r) and log (r)
corresponding to different values of embedding
dimensions (m), is plotted for every time series
and the slope gives the value for Correlation
dimension (d).
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? Generally, with increasing the value of
embedding dimension, the value of fractal
dimension should also increase until it reaches a
saturation value , this value will tell us the
saturation Correlation dimension and
corresponding saturation embedding dimension.
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? We find that there exist a saturation
correlation dimension (fractal dimension) only
for SO Index and PDO Index time series data. ?
From the figures, this is very clear that we can
not get the saturation value of Correlation
dimension for Antarctic Oscillation Index and
North Atlantic Oscillation Index. Thus we can say
that there exists no strange attractor for
Antarctic Oscillation.
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• The main objective of our study is to look for
  the possible teleconnection between the Southern
  Oscillation Index and Antarctic Oscillation
  Index.
• For this we surmise that if we can get the
  fractal dimensionality of nearly same value for
  SO and AO then they might said to be governed by
  the same strange attractor.
• Unfortunately here, we are not able to find the
  fractal dimensionality for AO. Similar is the
  case with NAO in northern hemisphere.

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? For SOI and NPOI, we get the minimum number and
maximum number of variables needed to model the
dynamics of the system, which are given
respectively by the integer next to saturation
value of Correlation dimension and by embedding
dimension. ? We find the value of dsat for SOI
is 7.28 and msat is 11 , and hence minimum number
and maximum number of variables needed to
describe the system are 8 and 11 . Similarly for
NPOI , dsat is 10.12 corresponding to msat 15
and hence minimum number and maximum number of
variables needed to describe the dynamics of NPOI
are 11 and 15.
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? This is not surprising, because the time series
used here includes weather phenomena, the long
range processes from season to season and the
interannual and interdecadal variability. A
dynamical system, which includes all these
aspects, depends on a large number of independent
variables (degree of freedom) and the inherent
noise. ? The fact that the attractor has a
fractal dimensionality provides a natural
explanation of the intrinsic variability of the
climate system despite its deterministic
character.
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? We got a clear indication that the attractor is
sufficiently embedded and has reached its
limiting dimension for SOI and NPOI time series.
? The possible cause behind the non-saturation
of correlation dimension in other time series may
be due to the small length of time series data /
this may also be due to random noise present in
the system (measurement).
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Conclusions
? Attractor reconstruction technique is used here
for single variable time series analysis and we
have calculated Correlation dimension for
different time series. ? Optimised box assisted
algorithm is used here to calculate Correlation
dimension which allows us to go for more number
of iterations as it takes less computation time
compared to GP algorithm. ? Minimum number and
maximum number of variables needed to describe
the dynamics of the system are obtained for SOI
and NPOI time series.
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? Although we are not able to establish the
teleconnection between SO and AO but one thing
which is very clear from our observations is that
AO plays the same role in Southern Hemisphere as
does NAO in Northern Hemisphere. Similar is the
case with PDO / NPO and SO because we get the
fractal dimensionality in the case of NPO and SO
and no such dimensionality is observed in the
case of AO and NPO.
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? The probable cause behind the non-observance of
strange attractor for Antarctica Oscillation
Index (AOI) may be the data scarcity in that
region and the problems also exist with the
length of available records. ? The need
therefore is of development of improved
techniques for estimating important geophysical
parameters from satellite and insitu data
observations, and of more number of data stations
in the Antarctic region and then possibly we can
get the desired tele-connection.
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? We hope that the technique used here may serve
as an additional tool for finding the
teleconnection between ENSO and AO and thus this
study can be used to understand the natural
climate variability in Antarctic region.
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THANK YOU