1. 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)

2. 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 !!

3. Background(2) ● Relation between dynamics of high-dim. chaos(GCTM) and dynamics of distribution(NLPF), observed in the study on Non-trivial corrective behavior Numerically observed behavior of NLPF system results of “linear stability analysis” for the dynamics of distribution (non-linear PF) Numerically observed behavior of GCM system (Paradoxical confliction in some cases)

4. ● Introduction:working 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

5. WORKING MODEL(1): Globally coupled tent map(GCTM) system ● N-dimensional map system, given by : parameter For, 2^n-band chaos appears in a(1-K)>1: expanding in all direction(N-dimensional chaos) (a>1) and (a(1-k)<1) : 1-dimensional chaos

6. 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)

7. Long transient in GCTM: Relation between GCTM and NLPF 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 Attractor of NLPF with piecewise constant distribution Limit of the sequence of attractor/natural invariant measure ??? naive expectation: the macroscopic property of infinitely large GCTM is well described with NLPF with absolutely continuous (piecewise constant) distribution. What kind of relation? (Large N limit)

8. A prominent discrepancy: Crisis occurs at a=2 or not ● GCTM with a=2 is critical (on the crisis bifurcation) for any N and k : (bounded attractor inevitably contain 1-cluster state) ● NLPF with a=2, ( ) 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

9. Numerical results: Long 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!] Inside of the “gap” Near the crisis line of NLPF

10. Phase diagram for GCTM S1: 1-dim bounded attractor SN: N-dim bounded attractor QS: No bounded attractor lifetime of transient diverges as. QQ: No bounded attractor (with possibly fairly long transient) a k

11. System size vs Lifetime System size Life time (in log scale) 10 10^7

12. System size vs Lifetime System size (log scale) Life time (log scale) 10 10^6 10000 10

13. 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.

14. An example of phenomena related to a variant of “quasi-stable”phase: 2-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

15. Bifurcation Diagram of single tent map ２ band ４ band a x

16. 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 (t=2n+1,any n in Z) it visits [0.5,1] segment At even time (t=2n,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

17. 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.

18. A:point cluster attractor (No 2-band state) B:stable 2-band state (both in NLPF & in GCTM) C:“quasi-stable” (in NLPF: stable) (in GCTM: unsbale) D:unstable (in both sys.) E: (in NLPF: No 2 band state) (GCTM: unstable) Numerically obtained Stability diagram for evenly partitioned 2-band states

19. Stability of 2-band states with biased partition The area in (a,k)-space gets smaller as the difference in band weight gets larger. GCTMNLPF If 2-band state with given weight ratio is stable in GCTM it is also stable in NLPF. GCTM NLPF

20. Parameter region with Stable 2-band state with various partition ratio(GCTM) a k (Outermost one corresponds 0.5:0.5, weight changed in step 0.1)

21. Parameter region with Stable 2-band state with various partition ratio(NLPF) a k (Outermost one corresponds 0.5:0.5, weight changed in step 0.1)

22. 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

24. 1-point cluster （ click to show life time distribution on each parameter set ）

25. 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 w linear 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 w linear/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

26. a=1.7,k=0.20 a=1.7,k=0.22 a=1.7,k=0.26 a=1.9,k=0.28 a=1.9,k=0.34 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

27. a=1.7,k=0.20 a=1.9,k=0.26 a=1.7,k=0.26 a=1.9,k=0.30 a=1.9,k=0.34 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

28. 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

35. a=1.7,k=0.20

36. a=1.7,k=0.22

37. a=1.7,k=0.26

38. a=1.9,k=0.28

39. a=1.9,k=0.34

40. a=1.7,k=0.20

41. a=1.7,k=0.26

42. a=1.9,k=0.26

43. a=1.9,k=0.30

44. a=1.9,k=0.34