Periodicity Manifestations in Turbulent Coupled Map Lattice 明明明明明明明 明明明明 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
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Periodicity Manifestations in Turbulent Coupled Map Lattice
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. - PowerPoint PPT Presentation
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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 NumbersK. Kaneko, Phys. Rev. Lett. 65, 1391, 1990.
Periodicity ManifestationsT. 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 ManifestationsT. Shimada, S. Tsukada, Physica D, 168-169, 126-135 ,2002.
T. Shimada, S. Tsukada, Prog. Theor. Phys. 108, 25,2002.
Phase SynchronizationH. Fujigaki, M. Nishi and T. Shimada, Phys. Rev. E53, 3192,1996H. Fujigaki and T. Shimada, Phys. Rev. E55, 2426, 1997.
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.
Maximal Lyapunov Exponents of p3c3MSCA events
Lyapunov Exponents and MSD
GCML a=1.90
Analytic Prediction at Maximal Population Symmetry
そこで, GCML の発展方程式
*
*
( ) ( ) ( ( )) ,
( ) ( ) ( ), , , , .
i i
i
x t f x t
a x th i N
h
2
1 1
1 1 12
は線形変換
*( ) ( ) ( )i iy t h x t 11
*( ) ( ) ( )a b ar a h a 21 1 1 1
に同値である.ただし,非線形性は,
そこで, GCML の発展方程式
*
*
( ) ( ) ( ( )) ,
( ) ( ) ( ), , , , .
i i
i
x t f x t
a x th i N
h
2
1 1
1 1 12
そこで, GCML の発展方程式
は線形変換
*( ) ( ) ( )i iy t h x t 11
( ) ( ), , ,i iy t by t i N 21 1 1
に同値である.ただし,非線形性は,
*
*
( ) ( ) ( ( )) ,
( ) ( ) ( ), , , , .
i i
i
x t f x t h
h a x t i N
2
1 1
1 1 12
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
の下で
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, )
h*
MSCA
*( ) ( ) ( )11i iy t h x t
t
X
t
X
t
X
t
y
t
y* (b)
single logistic map y(t) with b *( ) ( ) .1 1 1 1
br h
a
* * *( ) 11y h h
h* 消去
b a =b/r
* **( ) ( )
( ( ))2
1 12 2
ry b ry br y b
r
r をパラメータとした (a, e) 平面上の曲線
Foliation Curve of Window Dynamics
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… .
( )2 1MSD
T
t T
t
h h t h
Foliation Curves と平均場の2乗分散
Periodicity Manifestations and Statistics of Mean Field Time Series
p5c5 r=0.98
p5c3 r=0.98
0.94
0.94
h(t) distributions
p3c3 r=0.93
p3c2 r=0.92
GCML MSD a=1.90 and h(t) distributions
At MSD peak,
Double Gaussian.
At MSD valley,
simple Gaussian
with enhanced MSD.
(a), (b) The MSD curves of GCML along fixed r lines. (a) r=0.99, (b) r=0.95.
(c) Lyapunov exponent of a logistic map versus b measured with inclement b=10-4.
Fixed r-line に沿ってPMをみる.
Non-locally Coupled Map Lattices
( 1) (1 ) ( ( )) ( ), .P P Px t f x t h t P
Local mean field. ( ) ( ( )).P PQ Q
Q
h t W f x t
(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 .
(1 ) ( )PQ PQ PQ PQW c w with GCML-Limit
00 0
max
1/ ( 0)
( ) Exp( ( 1) / ( )
( ) ( )
POW
w EXP
CML
For GCML c=1/ N and w(ρ )=1.
MSD surfaces and their sections in D=1,2,3 POW
MSD surfaces for three non-local CMLs.
1.PQ PQW W
N
( ) ( ) ,PQP QQ
h t W f x t
( ) ( ) ( )P Pf x t f x t h t
22 ( ) ( )P Ph t f x t F
2 2 1( ) ( ) ,PQ PQ
Q Q
W WN
F
1/ 1/ 1/K N K F
( ) ( ) ( )P Ph t h t h t
A Working HypothesisGCML では , maps は平均場 h(t) に focus させられるのに対して ,
Measured ratio h(hP Ä h)2iÉ=h(f P Ä h)2iÉ (averaged
over 100 steps) versus F (ã ) in POWã . ã inclemented
by 0:5 between 0:5Ä 8:0; D = 1Ä 3. " is set at 0.02,
0.08, 0.0352(p6c6), 0.045(p3c2) for (a)-(d).
Time-dependence testTest over α and D. ( each run averaged 100 steps.)
仮定仮定 22 のテスト のテスト (POW-Model)(POW-Model)
DÄf(xP (t)) Ä h
Å2E
É
úêhP (t) Ä h
ë2ù
É
F(a) (b)
(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 non-locally 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?