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 /