スライド 1

Periodicity Manifestations in Turbulent
Coupled Map Lattice
明治大理工物理
島田徳三
1．A brief introduction to GCML.
2. Formation of Periodic Clusters in the Turbulent GCML.
Foliation of Periodic Windows of Element Maps.
3. Universality in Periodicity Manifestations.
4. Discussions
GCML :Phase Diagram
K. Kaneko, Phys. Rev. Lett. 63, 219, 1989.
GCML:Law of Large Numbers
K. Kaneko, Phys. Rev. Lett. 65, 1391, 1990.
Periodicity Manifestations
T. Shibata and K. Kaneko, Physica D124, 177,1998.
T. Shimada and K. Kikuchi, Phys. Rev. E 62, 3489, 2000.
A. Parravano and M. G. Cosenza, Int. J. Bifurcation Chaos 9, 2331,1999.
Universality in Periodicity Manifestations
T. Shimada, S. Tsukada, Physica D, 168-169, 126-135 ,2002.
T. Shimada, S. Tsukada, Prog. Theor. Phys. 108, 25,2002.
Phase Synchronization
H. Fujigaki, M. Nishi and T. Shimada, Phys. Rev. E53, 3192,1996
H. Fujigaki and T. Shimada, Phys. Rev. E55, 2426, 1997.
Globally Coupled Map Lattice
• 全部でN箇の写像素子を平均場を通して結合させ,平均
化の相互作用のもとで発展させる．
GCMLの相図
Random motion in a
unity
Coherence

Curve of Balance
a
Randomness
Periodicity
Manifestations in
Chaos
MaximallySymmetricClusterAttractors
（MSCA)
 X1   X 2 
X   
X3
2 


 

  

  
X
 p   X1 
 Xp 
 X 
1 






X
 p1 
系の素子全体が自発的に形成する集団周期運動状態．
ｐ3 ＭＳＣＡでは，素子は３つのクラスターに同数ずつ
分かれ，相対的に位相が２π／３ずれた）周期３運動をす
る．平均場の値が一定なので，系は安定性を持つ．
Fortran executable files to see typical PMs
are uploaded at the entrance to this PPT
show in the Shimada’s page.
Please download them and try.
p5c3
p3c2
p3c3MSCA
Maximal Lyapunov Exponents of p3c3MSCA events
Lyapunov Exponents and MSD
GCML a=1.90
Analytic Prediction at
Maximal Population
Symmetry
ＭＳＣＡ状態では素子たちの平均場 h(t) が時間に依存しない
定数 h* になる．
そこで，GCMLの発展方程式
xi (t  1)  (1   ) f ( xi (t ))   h* ,
 (1     h* )  a(1   ) xi2 (t ), i  1, 2, , N .
は線形変換
yi (t )  (1     h* )1 xi (t ) の下で
yi (t  1)  1  byi 2 (t ), i  1, , N
に同値である．ただし，非線形性は，
a  b  ar  a(1   ) 1   (1  h* )   a(1   ) 2
とさがっている．この b の値は,単一素子のp周期窓のパラメー
ター区間に含まれなければならない.
GCML (a, ） MSCA
X
ｈ＊
ｔ
yi (t )  (1     h* )1 xi (t )
single logistic map y(t) with b
r


b
 (1   ) 1   (1  h* )  1.
a
h* 消去
ｙ
ｙ＊(b)
ｔ
y*  (1     h* )1 h*
Foliation Curve of Window Dynamics
r
b
a =b/r
 1
 ry (b) 
ry (b)
 r (1  y* (b))  

2
 2 
*
*
2
rをパラメータとした(a, e)平面上の曲線
Foliation Curves と平均場の２乗分散
MSD   h 2 
Foliation curves from outstanding windows with ｐ = 7, 5, 7, 13, 8, 3, 5, 4 with increasing b．
(A: intermittency, B: lower threshold, C: the first bifurcation, D: closing point).
The expected zones of onset of the window dynamics are shown in the panels at a=1.8, 1.9, 2.0.
The dashed line is the boundary curve from the band merging point (m) at b=1.543689… ．

1 t T
 h(t )  h
T t

Periodicity Manifestations and Statistics of Mean Field Time Series
h(t) distributions
p3c3 r=0.93
p3c2 r=0.92
p5c5 r=0.98
0.94
p5c3 r=0.98
0.94
GCML MSD a=1.90 and h(t) distributions
At MSD peak,
Double Gaussian.
At MSD valley,
simple Gaussian
with enhanced MSD.
Fixed r-line に沿ってＰＭをみる．
(a), (b) The MSD curves of GCML along fixed r lines. (a) r=0.99, (b) r=0.95.
-4.
(c) Lyapunov exponent of a logistic map versus b measured with inclement b=10
Non-locally Coupled Map Lattices
xP (t  1)  (1   ) f ( xP (t ))  hP (t ), P .
Local mean field.
hP (t )   WPQ f ( xQ (t )).
Q
(An weighted average of map values around P ).
GCML:
No concept of distance. Zero dim.
f(xi) s are uniformly pulled to the system
mean filed h(t) by a factor 1  1   .
CML:
f(xP) at site P is pulled to the local mean field
hP(t) by 1  1   .
WPQ  c  PQ  (1   PQ ) w(  PQ ) 
with

1/  
POW

w(  )   Exp((   1) / 0 EXP0

 (   )
CML

GCML-Limit
(  0)
(  0  )
(   max )
MSD surfaces and their sections in D=1,2,3 POW
MSD surfaces for three non-local CMLs.
A Working Hypothesis
GCMLでは , maps は平均場 h(t)にfocusさせられるのに対して,
CMLでは, mapはそれぞれの位置での局所平均場hP(t)にfocusする．
そこで, Periodicity Manifestationsの強度は,
 hP (t )  hP (t )  h(t )
の2乗分散で決まり,この分散が等しい場合は同じ強さでPMが起こると仮定する．
W PQ  WPQ 
1
.
N
 hP (t )   W PQ  f  xQ (t ) ,
但し,  f  xP (t )   f  xP (t )   h(t )
Q
ここで第2の仮定として、CMLのmapは各時刻 t で, 空間的な相関を持たないとする．
そうすれば、重み付け平均に対する大数の法則から,
分散の評価
 hP2 (t )
F


F
 (W
PQ
Q
 f  x
(t )  
 (W
)2 
)2 
P
PQ
Q
を得る．特にＣＭＬκでは、rangeκ内の素子数をKとして
F  1/ K 1/ N  1/ K
2

1
,
N
仮定2のテスト (POW-Model)
Time-dependence test
Test over α and D.
（each run averaged 100 steps.)
DÄ
Å2 E
f (x P (t )) Ä h
É
(b)
(a)
F
úê
Measured rat io h(hP Ä h) 2 i É =h(f P Ä h) 2 i É (averaged
over 100 st eps) versus F ( ã ) in POW ã . ã inclement ed
by 0:5 between 0:5 Ä 8:0; D = 1 Ä 3. " is set at 0.02,
0.08, 0.0352(p6c6), 0.045(p3c2) for (a)-(d).
h P (t ) Ä h
ë2 ù
É
(c)
(d)
Test of the Hypothesis in POWα
Test of the Hypothesis over three CMLs
Predicting PMs from D=1 POW only.
Conclusions, Questions, Discussions.
1. We have found that coupled chaotic maps under mean field interaction
reduce the nonlinearity and form periodic cluster attractors.
2. There is a universality in the periodicity manifestations in three nonlocally coupled map lattices. The controlling factor is the variation of the
local mean field around the system mean field.
3. Why Nthreshold , rthreshold ?
cf. SSB in Field Theory.
4. Map and Flow Correspondence.
(Logistic map vs Duffine Oscillators )
Coupled (quantum) kicked rotators?
Some Comments Follows:
N
100
1000
10000
100000
1000000
GCML a=1.90, =0.0682.
Synchronization and Metamorphosis
2つのローレンツアトラクターの双方向結合系．一方は周期領域(r= 300), 他方はカオス領域(r= 28)
のパラメータを与えている (b= 8/ 3, P= 10は共通). 結合比を連続的に変化させると２つの流れ素子は,
位相同期を保ちつつ, 周期軌道(上部) からカオス軌道(下部) へ連続遷移をする。パソコンのディスプレ
イ上で, カオス, 周期をわたる素子たちのダンスが見える。 [藤垣+ ＴＳ Phys. Rev. E53, E55].
 (r=28) x (1-) (r=300)
=0
=1
ＧＣＭＬ of 50 Duffine Oscillators
Two Cluster Regime
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