Long time correlation due to high-dimensional chaos

in globally coupled tent map system

Tsuyoshi ChawanyaTsuyoshi Chawanya

Department of Pure and Applied Mathematics,Department of Pure and Applied Mathematics,Graduate School of Information Science and Technology,Graduate School of Information Science and Technology,

Osaka UniversityOsaka University

(PFDW05)(PFDW05)

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 DISTRIBUTIONinstead of N(many)-INDIVIDUAL

VARIABLES

“Physically” natural approach !!

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 observedbehavior of NLPF system

results of “linear stability analysis”for the dynamics of distribution

(non-linear PF)

Numerically observedbehavior of GCM system

N

(Paradoxical confliction in some cases)

● Introduction:working modelsglobally coupled tent map system(GCTM)non-linear Perron Frobenius system

● “Long transient” in high-dimensional GCTMas a shadow of macroscopic (NLPF)

attractor(simple version)

● Apparent power-law distribution in 2-band intermittency as a derivative of the longtransient mechanism

WORKING MODEL(1):

Globally coupled tent map(GCTM) system

● N-dimensional map system, given by

: parameter

x i ,t 1 1 k a12

xi ,t12

akN j 1

N12

x j ,t12

For , 2^n-band chaos appears in k 0 2 2 n 1

a 2 2 na(1-K)>1: expanding in all direction(N-dimensional chaos)

(a>1) and (a(1-k)<1) : 1-dimensional chaos

a , k , N

WORKING MODEL (2):

Non-linear Perron Frobenius(NLPF) system

● Dynamical system of “distribution” ,

: parameter

corresponds to “one-body distribution” of GCTM

t 1 x f y , t x t y dy ,

f x , t 1 k a12

x12

kh t ,

h t a12

x12 t x dx

Good point:GCTM with different system size(N) can be handledin the same phase space.(by using correspendence betweenN-dimensional GCTM system

and NLPF with Phase space restricted on sum of N deltas)

x

x

a , k

Long transient in GCTM:Relation between GCTM and

NLPFnaive 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 behaviorof GCTM with N elements

Attractor of NLPFwith piecewise

constant distribution

Limit of the sequenceof attractor/naturalinvariant 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)(Large N limit)

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)

0 k 1 2

Good motivation for the investigation on the behaver of “large dimensional” system

with parameter set in the space between these two lines

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

Phase diagram for GCTM

S1: 1-dim bounded attractor

SN: N-dim bounded attractor

QS: No bounded attractor lifetime of transient diverges as .N

QQ: No bounded attractor (with possibly fairly long transient)

a

k

System size vs Lifetime

System size

Life tim

e (in log scale)

10

10^7

System size vs Lifetime

System size (log scale)

Life tim

e (log scale)

10

10^6

1000010

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”

log N

In high-dimensional GCTM system, the Attractor vanishes quite slowly.

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

Bifurcation Diagram of single tent map

２ band４ band

a

x

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] segmentAt 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

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.

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 diagramfor evenly partitioned 2-band states

Stability of 2-band states with biased partition

The area in (a,k)-space gets smaller as the difference in band weight gets larger. GCTM NLPF

If 2-band state with given weight ratio is stable in GCTMit is also stable in NLPF.

GCTM NLPF

Parameter region with Stable 2-band statewith various partition ratio(GCTM)

a

k

(Outermost one corresponds 0.5:0.5, weight changed in step 0.1)

Parameter region with Stable 2-band statewith various partition ratio(NLPF)

a

k

(Outermost one corresponds 0.5:0.5, weight changed in step 0.1)

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

1-point cluster

（ click to show life time distribution on each parameter set）

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 wlinear 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 wlinear/no dependence of log(life time) on N?

lifetime distributionfor fixed a,k,w

Lifetime vs w,N

Reinjection frequency

(2) and (3) leads to emergence of the power-law range in life-time distributionwidth of the power-law range (in log-scale) that grows linearly with N

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 statesas a function of band weight ratio

horizontal-axis: band weight ratiovertical-axis: life time ( in log scale)

plotted for a certain values of Na and k : fixed

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 distributionof band weight ratio

horizontal-axis: band weight ratiovertical-axis: frequency( in log scale)

plotted for a certain values of Na and k : fixed

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

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

a=1.7,k=0.20

a=1.7,k=0.26

a=1.9,k=0.26

a=1.9,k=0.30

a=1.9,k=0.34