Incidence of q-statistics at the transitions to chaos
Alberto Robledo
Dynamical Systems andStatistical MechanicsDurham Symposium3rd - 13th July 2006
• How much was known, say, ten years ago?
• Fluctuating dynamics at the onset of chaos (all routes)
Brief answers are given in the following slides
• Which are the relevant recent advances? Is the dynamics fully understood now?
• Is there rigorous, sensible, proof of incidence of q-statistics at the transitions to chaos?
Subject:
• What is q-statistics for critical attractors?
• What is the usefulness of q-statistics for this problem?
• What is the relationship between q-statistics and the thermodynamic formalism?
Questions addressed:
Ten years ago
• Numerical evidence of fluctuating dynamics (Grassberger & Scheunert)
• Adaptation of thermodymamic formalism to onset of chaos (Anania & Politi, Mori et al)
• But… implied anomalous statistics overlooked
t
idxi
dx
i1ln
1
t
idxi
dx
t i1ln
ln
1
1
t
idxi
dx
t i1ln
1
1
Dynamics at the ‘golden-mean’ quasiperiodic attractor
Red: Black: Blue:
Thermodynamic approach for attractor dynamics(Mori and colleagues ~1989)
• ‘Special’ Lyapunov coefficients
• Partition function
• Free energies
• Equation of state and susceptibility
,1,)(
lnln
1),(
1
00
t
i i
i tdx
xdf
txt );0();( )( tPttP
ttWtWtPdtZ ),(,),();()q,( q1
)1q()q()( ,ln
)q,(lnlim)q(
t
tZt
q
)q()q(,
q
)q()q(
d
d
d
d
q ~ ‘magnetic field’ ~ ‘magnetization’
Mori’s q-phase transition at the period doubling onset of chaos
• Is theTsallis index q the value of q at the Mori transition?
Static spectrum Dynamic spectrum
q)1q()q(
,q)()q(
D
f
)1q()()q(
),;0();( )( tPttP
1qm
)(
75.0
q
Mori’s q-phase transition at the quasiperiodic onset of chaos
• Is theTsallis index q the value of q at the Mori transition?
Static spectrum Dynamic spectrum
q)1q()q(
,q)()q(
D
f
)1q()()q(
),;0();( )( tPttP
)(
1qm95.0
Today
• [Rigorous, analytical] results for the three routes to chaos (e.g. sensitivity to initial conditions)
• Hierarchy of dynamical q-phase transitions
• q values determined from theoretical arguments
• Temporal extensivity of q-entropy
• Link between thermodymamics and q-statistics
Trajectory on the ‘golden-mean’ quasiperiodic attractor
1mod),2sin(2
11 ttt
Trajectory on the Feigenbaum attractor
11,1)( 21 tttt xxxfx
q-statistics for critical attractors
Sensitivity to initial conditions
• Ordinary statistics: • q statistics:0
00
0
lim),(x
xtx t
x
])(exp[),( 010 txtx ])([exp),( 00 txtx qq
• q-exponential function: qq xqx 1
1
])1(1[)(exp
)(explim)(exp1
xx qq
• Basic properties:
qQxx Qq 2,)]([exp)(exp 1
qqq xdxxd )]([exp)(exp
BAbxaxfx ttt ,),(1
0x )t(independent of for (dependent on 0x for all t )
q-exponential function
Entropic expression for Lyapunov coefficient
• Ordinary statistics: • q statistics:
])0()([1
lim qqt
q StSt
ii
i ppS ln1
• q-logarithmic function: );(1
1)(ln
1
RqRyq
yy
q
q
)(lnlim)ln(1
yy qq
• Basic properties:
qQyy Qq 2,)/1(ln)(ln
xxx qqqq ))((lnexp))((expln
iQi
iiqi
qiq ppppS lnln or
Analytical results for the sensitivity
Power laws, q-exponentials and two-time scaling
,2ln)12(
ln,
ln
2ln1,exp)( )()(
0
lqtx l
qlqqt
1,)( ink
int xx
2ln/ln
121
l
tk
12,12)12(,/exp)( )0(0 ltltttx w
kwqqt
,...1,0,...,2,1),12)(12( lklt k
Sensitivity to initial conditions within the Feigenbaum attractor
• Starting at the most crowded (x=1) and finishing at the most sparse (x=0) region of the attractor
• Starting at the most sparse (x=0) and finishing at the most crowded (x=1) region of the attractor
,2ln)12(
ln)1(,
ln)1(
2ln1,exp)( )()(
0
l
z
zqtx l
qlqqt
,2ln)12(
ln)1(2,
ln)1(
2ln12,exp)( )(
2)(
220
l
z
zqtx l
qlqqt
,...1,0,...,1,0,12)12( kllt k
,...1,0,...,1,0,12)12( kllt k
Sensitivity to initial conditions within the golden-mean quasiperiodic attractor
• Starting at the most sparse (θ=0) and finishing at the most crowded (θ= ) region of the attractor
• Starting at the most crowded (θ= ) and finishing at the most sparse (θ=0) region of the attractor
gmgm
gmlkq
gm
gmlkqqt wlwlk
wqt
ln)(
ln2,
ln2
ln1,exp)(
2),(),(
0
gmgmgm
gmlkq
gm
gmlkqqt wwlwlk
wqt
ln)(
ln2,
ln2
ln12,exp)(
2),(),(
220
,...1,0,...,2,1,...,2,1,1)( 2 mlkmFFmlt kk
,...1,0,...,2,1,...,2,1,1)( 2 mlkmFFmlt kk
Thermodynamic approach and q-statistics
,)(
lnln
1),(
)(
in
int
in dx
xdg
txt
,2ln)12(
ln)(ln
1)1,( )(
1
)(lq
xin
int
qin ldx
xdg
txt
in
,...1,0,...,1,0,12)12( lklt k
Mori’s definition for Lyapunov coefficient at onset of chaos
is equivalent to that of same quantity in q-statistics
Dynamic spectrum
7555.0ln
2ln1
qm2445.0
ln
2ln1
q
Two-scale Mori’s λ(q) and (λ) for period-doubling threshold
3236.12ln
ln
q
),;0();( )( tPttP
Dynamic spectrum
949.0ln
ln1
gm
gmwqm
0510.0
ln2
ln1
gm
gmwq
Two-scale Mori’s λ(q) and (λ) for golden-mean threshold
0537.1ln
ln2
gm
gmq w
),;0();( )( tPttP
11 ,2,
,
,1 or,)(
nn Fmmmnmmmn
mn
mnn dxxd
d
dm
nnn
n F
my
mymy
or
2),(lim)(
1or12,)(
)1()(
,
,
n
n
n
n
n
mn
tmnt Ftt
m
m
d
dm
Trajectory scaling function σ(y) → sensitivity ξ(t)
Hierarchical family of q-phase transitions
,...,2,1,,...,1,0,,)( 11
JJjayay jjj
Spectrum of q-Lyapunov coefficients with common index q
• Successive approximations to σ(y),
lead to:
and similarly with Q=2-q
;ln)(
)/ln(,
)/ln(
ln1
or,,2ln)12(
)/ln(,
)/ln(
2ln1
where,exp)(
2
1),(
1
1)(
1
)(
10
gmgm
jjlkq
jj
gm
jjkq
jj
kqq
n
j
jt
wlwlk
wq
kq
tx
q
)q()q(
d
d )1q()q()(
)(y
nAmy /
Infinite family of q-phase transitions
• Each discontinuity in σ(y) leads to a couple of q-phase transitions
q2
)1(q
)1(2 q
q )1(2 q )1(
q
1qm
qm 1
Temporal extensivity of the q-entropy
)(exp)(),(),( )()( ttWtP lqq
lq
)1q)/(1()(q1)( )1(1),()q,(-ql
qlq tqtWtZ
),(q1(1)()q,( q
q
1
t)StptZW
ii
therefore
and
Precise knowledge of dynamics implies that
,)( )( ttS lqq When q=q
with
.lnand allfor )()( qq1 WSitWtpi
,...1,0fixedwith ,...,2,1),12)(12( lklt k
Linear growth of Sq
)(ln tt qq
)(tStK qq
• “Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points”, A. Robledo, Physica A (in press) & cond-mat/0606334
Where to find our statements and results explained
• “Critical attractors and q-statistics”, A. Robledo, Europhys. News, 36, 214 (2005)
Concluding remarks
• Usefulness of q-statistics at the transitions to chaos
q-statistics and the transitions to chaos
• The fluctuating dynamics on a critical multifractal attractor has been determined exactly (e.g. via the universal function σ)
• The entire dynamics consists of a family of q-phase transitions
• Tsallis’ q is the value that Mori’s field q takes at a q-phase transition
• A posteriori, comparison has been made with Mori’s and Tsallis’ formalisms
It was found that:
• The structure of the sensitivity is a two-time q-exponential
• There is a discrete set of q values determined by universal constants
• The entropy Sq grows linearly with time when q= q
Onset of chaos in nonlinear maps
Critical clusters Glass formation
Localization
Intermittency route
Quasiperiodicity route
Period-doubling route
t
idxi
dx
tE
i1ln
1
t
idxi
dx
tE
i1ln
1
t
idxi
dx
tE
i1ln
1
t
idxi
dx
tE
i1ln
1
t
idxi
dx
tE
i1ln
1
t
idxi
dx
tE
i1ln
1
Feigenbaum’s trajectory scaling function σ(y)
1
12/ nmy
1
mnmn dd ,,1 /
Trajectory scaling function σ(y) for golden mean threshold
)(y
nFmy /
mnmn dd ,,1 /