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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.

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Periodicity Manifestations in Turbulent Coupled Map Lattice

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  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

  2. 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.

  3. 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．

  4. Coherence Curveof Balance Periodicity ManifestationsinChaos GCMLの相図 Random motion in a unity Randomness

  5. MaximallySymmetricClusterAttractors（MSCA) 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．

  6. 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

  7. Maximal Lyapunov Exponents ofp3c3MSCA events

  8. Lyapunov Exponents and MSD GCML a=1.90 Analytic Prediction at Maximal Population Symmetry

  9. 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 　そこで，GCMLの発展方程式 　そこで，GCMLの発展方程式 　そこで，GCMLの発展方程式 は線形変換 は線形変換 の下で に同値である．ただし，非線形性は， に同値である．ただし，非線形性は， 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.

  10. X X X ｙ ｔ ｔ ｔ ｔ ｔ GCML (a, e） MSCA ｈ＊ single logistic map y(t) with b h* 消去 ｙ＊(b)

  11. Foliation Curve of Window Dynamics r a =b/r b rをパラメータとした(a, e)平面上の曲線

  12. Foliation Curves と平均場の２乗分散 Foliation curves from outstanding windows withｐ = 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… ．

  13. 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

  14. GCML MSD a=1.90 and h(t) distributions At MSD peak, Double Gaussian. At MSD valley, simple Gaussian with enhanced MSD.

  15. 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. (c) Lyapunov exponent of a logistic map versus b measured with inclement b=10-4.

  16. MSD surfaces and their sections in D=1,2,3 POW

  17. MSD surfaces for three non-local CMLs.

  18. 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, 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． 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, 分散の評価 を得る．特にＣＭＬκでは、rangeκ内の素子数をKとして .

  19. 仮定2のテスト (POW-Model) Time-dependence test Test over α and D. （each run averaged 100 steps.) (b) (a) (c) (d)

  20. Test of the Hypothesis in POWα

  21. Test of the Hypothesis over three CMLs

  22. Predicting PMs from D=1 POW only.

  23. 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:

  24. N 100 1000 10000 100000 1000000

  25. GCML a=1.90, e=0.0682.

  26. Synchronization and Metamorphosis q (r=28) x (1-q) (r=300) q=0 q=1

  27. ＧＣＭＬ of 50 Duffine Oscillators Two Cluster Regime

  28. Bye for now!!! Click here to go back to top of the how. Click here to close the show.

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