Long time correlation due to highdimensional chaos in globally coupled tent map system

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Long time correlation due to high-dimensional
chaos in globally coupled tent map system

• Tsuyoshi Chawanya
• Department of Pure and Applied Mathematics,
• Graduate School of Information Science and
  Technology,
• Osaka University
• (PFDW05)

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Background(1)

• Subject of study Macroscopic behavior in
  extensively high-dimensional systems How can
  we analyze them?
• Target phenomenon Why are the intermittent
  phenomena and/or long transient behavior observed
  so ubiquitously in high-dimensional chaotic
  systems?
• Approach Concentrate on DISTRIBUTION instead of
  N(many)-INDIVIDUAL VARIABLES Physically
  natural approach !!

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Background(2)

• Relation between dynamics of high-dim.
  chaos(GCTM) and dynamics of distribution(NLPF),
  observed in the study on Non-trivial corrective
  behavior

results of linear stability analysis for the
dynamics of distribution (non-linear PF)
Numerically observed behavior of NLPF system
(Paradoxical confliction in some cases)
Numerically observed behavior of GCM system
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• Introductionworking models globally coupled
  tent map system(GCTM) non-linear Perron
  Frobenius system
• Long transient in high-dimensional GCTM as a
  shadow of macroscopic (NLPF) attractor (simple
  version)
• Apparent power-law distribution in 2-band
  intermittency as a derivative of the
  longtransient mechanism

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WORKING MODEL(1)Globally coupled tent
map(GCTM) system

• N-dimensional map system, given by
  parameter

(agt1) and (a(1-k)lt1) 1-dimensional chaos
a(1-K)gt1 expanding in all direction(N-dimensional
chaos)
For , 2n-band chaos appears in
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WORKING MODEL (2)Non-linear Perron
Frobenius(NLPF) system

• Dynamical system of distribution
  , parameter corresponds to
  one-body distribution of GCTM

Good point GCTM with different system size(N)
can be handled in the same phase space. (by
using correspendence betweenN-dimensional GCTM
system and NLPF with Phase space restricted on
sum of N deltas)
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Long transient in GCTMRelation between GCTM and
NLPF
naive expectation the macroscopic property of
infinitely large GCTM is well described with NLPF
with absolutely continuous (piecewise constant)
distribution.

• naive expectation
• the macroscopic property of infinitely large GCTM
  is well described with NLPF with absolutely
  continuous distribution (piecewise constant
  distribution).

attractor/natural invariant measure of NLPF with
N-delta distribution
asymptotic behavior of GCTM with N elements
(Large N limit)
(Large N limit)
???
Limit of the sequence of attractor/natural invaria
nt measure
Attractor of NLPF with piecewise constant
distribution
What kind of relation?
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A prominent discrepancyCrisis occurs at a2 or
not

• GCTM with a2 is critical (on the crisis
  bifurcation) for any N and k (bounded
  attractor inevitably contain 1-cluster state)
• NLPF with a2, ( ) with smooth
  distribution is not on the crisis (for initial
  states with bounded total variation, total
  variation never diverge)

Good motivation for the investigation on the
behaver of large dimensional system with
parameter set in the space between these two lines
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Numerical resultsLong transient (quasi-stable
phase)and Phase diagram

• Relation between lifetime of bounded state and
  system size
• Numerically obtained phase diagram wide
  discrepancy of crisis bifurcation line!

Near the crisis line of NLPF
Inside of the gap
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Phase diagram for GCTM
S1 1-dim bounded attractor
SN N-dim bounded attractor
QS No bounded attractor lifetime of transient
diverges as .
k
QQ No bounded attractor (with possibly fairly
long transient)
a
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System size vs Lifetime
107
Life time (in log scale)
10
System size
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System size vs Lifetime
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Life time (log scale)
10
10000
10
System size (log scale)
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Summary of this part

• Large discrepancy in the position of the crisis
  bifurcation line of GCTM and that of NLPF (with
  piecewise constant distribution function) is
  observed
• GCTM with parameter value inbetween these two
  bifurcation lines exhibits long transient
  behavior, whose life time grows with N as
• Consistent with the estimaion derived from the
  view as escape from macroscopic/thermodynamic
  attractor induced by noise due to finite size
  effect

In high-dimensional GCTM system, the Attractor
vanishes quite slowly.
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An example of phenomena related to a variant of
quasi-stablephase2-band intermittency

• 2-band states in Tent-map, GCTM and NLPF
• Phase diagram
• Observed life time distribution
• power-law (with index near -1)as a consequence
  of non-singular parameter dependence

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Bifurcation Diagram of single tent map
x
a
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Working definition for transient 2-band state

• Let us note a property of 2-band state in
  tent-map system an element in 2-band chaos
  takes one of the following 2-states, i.e. At
  odd time (t2n1,any n in Z) it visits 0.5,1
  segment At even time (t2n,any n in Z) it visits
  0.5,1 segment
• Working definition for Transient 2-band state
  let us divide the elements into two groups,
  depending on the last visit to 0,0.5 segment is
  odd-time or even-time. If the group does not
  change for a certain period, we will consider
  the system is in a (transient) 2-band state
• Working definition for (transient) 2-band state
  in NLPF odd-time image of the critical point
  (0.5) is in 0.5,1 segment

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On the stability of 2-band states

• The stability of 2-band states depends on a,k and
  the weight ratio of the 2 bands. (no direct
  dependence on N is observed in numerical
  calculation)
• The crysis (band merging) may occur at different
  point in GCTM and NLPF.

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Numerically obtained Stability diagram for evenly
partitioned 2-band states
Apoint cluster attractor (No 2-band
state) Bstable 2-band state (both in NLPF
in GCTM) Cquasi-stable (in NLPF stable)
(in GCTM unsbale) Dunstable (in both sys.) E
(in NLPF No 2 band state) (GCTM unstable)
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Stability of 2-band states with biased partition

• The area in (a,k)-space gets smaller as the
  difference in band weight gets larger.
  GCTM NLPF
• If 2-band state with given weight ratio is stable
  in GCTMit is also stable in NLPF.

GCTM
NLPF
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Parameter region with Stable 2-band state with
various partition ratio(GCTM)
(Outermost one corresponds 0.50.5, weight
changed in step 0.1)
k
a
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Parameter region with Stable 2-band state with
various partition ratio(NLPF)
(Outermost one corresponds 0.50.5, weight
changed in step 0.1)
k
a
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Numerical results observed intermittency and
its statistical properties

• Some examples of temporal sequence of mean-field
  h(t)
• Life-time distribution of temporal 2-band states

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1-point cluster
(click to show life time distribution on each
parameter set)
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On the mechanism of apparent power-law

• (1) life-time distribution with fixed a,k and
  band-weight ratio(w)each of them exhibits clear
  exponential decay
• (2) life-time as a function of band-weight ratio
  and N (for fixed a,k)smooth dependence on
  wlinear dependence of log(life time) on N
• (3) re-injection frequency as a function of
  band-weight ratio and N (for fixed a,k)smooth
  dependency on wlinear/no dependence of log(life
  time) on N?

lifetime distribution for fixed a,k,w
Lifetime vs w,N
Reinjection frequency
(2) and (3) leads to emergence of the power-law
range in life-time distribution width of the
power-law range (in log-scale) that grows
linearly with N
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a1.9,k0.34
a1.7,k0.26
a1.9,k0.28
a1.7,k0.22
Life time of 2-band states as a function of band
weight ratio horizontal-axis band weight
ratio vertical-axis life time ( in log
scale) plotted for a certain values of N a and k
fixed
a1.7,k0.20
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a1.9,k0.34
Frequencey distribution of band weight
ratio horizontal-axis band weight
ratio vertical-axis frequency( in log
scale) plotted for a certain values of N a and k
fixed
a1.9,k0.30
a1.7,k0.26
a1.9,k0.26
a1.7,k0.20
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Summary and discussion

• In GCTM sysytem with large but finite number of
  elements, itinerant behavior could appear as a
  shadow of multi-stability in associated Macro
  scopic (NLPF) dynamics
• Observed in wide area in parameter(a,k) space
• Life-time distribution exhibits power-law over a
  certain decade, and the width of the range grows
  with N

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a1.7,k0.20
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a1.7,k0.22
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a1.7,k0.26
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a1.9,k0.28
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a1.9,k0.34
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a1.7,k0.20
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a1.7,k0.26
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a1.9,k0.26
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a1.9,k0.30
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a1.9,k0.34