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.

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

Periodicity Manifestations 　in 　 Chaos

Random motion in a unity

a

Coherence

Curve of Balance

Randomness

MaximallySymmetricClusterAttractors（ MSCA)

p

p p

X XX

X XX

X XX

1 2

2 13

11

系の素子全体が自発的に形成する集団周期運動状態．ｐ 3 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２ π ／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．

p5c3 p3c2 p3c3MSCA

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 ,

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 ,

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．

分散の評価

但し ,

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 ,

を得る．特にＣＭＬ κ では、 rangeκ 内の素子数を K として

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?

Some Comments Follows:

N

100

1000

10000

100000

1000000

GCML a=1.90, =0.0682. 　

Synchronization and Metamorphosis

=0

=1

(r=28) x (1-) (r=300)

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].

ＧＣＭＬ of 50 Duffine Oscillators

Two Cluster Regime

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