Long time correlation due to high-dimensional chaos in globally coupled tent map system
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Transcript of Long time correlation due to high-dimensional chaos in globally coupled tent map system
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