Spatio-Temporal Chaos

and Defects

Hermann Riecke, Northwestern University

Yuan-Nan Young, CTR Stanford University

Glen Granzow, Lander UniversityCristian Huepe, Northwestern University

0 10 20 30 40 50 60y-Location

20000

20050

20100

20150

Tim

e

Dynamics Days 2003

supported by NSF, NASA, and DOE

Spatio-Temporal Chaos

Spiral-Defect Chaos

(Morris, Bodenschatz, Cannell, Ahlers, 1993)

T+dT

T

Undulation Chaos

(Daniels and Bodenschatz, 2002)

T+dT

T

up

down

Dislocations in Stripe-Based Spatio-temporal Chaos

• statistics for number of dislocations- complex Ginzburg-Landau equation (Gil, Lega, and Meunier, 1990)- electroconvection (Rehberg et al., 1989)- undulation chaos (Daniels and Bodenschatz, 2002)

• reconstruction of pattern from dislocations (La Porta and Surko, 1999)

• dynamical dimension of defects- complex Ginzburg-Landau equation (Egolf, 1998)- excitable media (Strain and Greenside, 1998)

Here:

• hexagonal patterns: penta-hepta defects

• stripe-based patterns: statistics of trajectories of dislocations

Non-Boussinesq Convection

(Bodenschatz, de Bruyn, Ahlers, Cannell, 1991)

T+dT

T

Ginzburg-Landau Equations for Hexagons

~

~

~ τ1τ2

n2

n1

n3 τ3

q

q2

3

1q

• Envelope equations:Slow variation of wavevector and magnitude

v(r̃, t̃) = ε3

∑j=1

A j(εr̃,ε2t̃)︸ ︷︷ ︸

∝exp(iεq j ·r̃)

eiq̃ j ·r f (z)+ c.c.+h.o.t.

∂tA1 = µA1 +(n1 ·∇)2A1 +A∗2A∗3 −A1|A1|

2

−(ν+ γ)A1|A2|2 − (ν− γ)A1|A3|2 − iβA1(τ̂1 ·∇)Q

+iα2 (A∗3 (τ2 ·∇)A∗2 −A

∗2 (τ3 ·∇)A

∗3)

+i(α1−α3)A∗3 (n2 ·∇)A∗2 + i(α1+α3)A

∗2 (n3 ·∇)A

∗3

∇2Q =3

∑j=1

[2(n̂ j ·∇)(τ̂ j ·∇)+ τ

((n̂ j ·∇)2 − (τ̂ j ·∇)2

)]|A j|

2

with drift velocity V ∝ ∇× (Qêz)

• Strong resonance A∗2A∗3 • Variational system

Ginzburg-Landau Equations for Hexagons

~

~

~ τ1τ2

n2

n1

n3 τ3

q

q2

3

1q

• Envelope equations:Slow variation of wavevector and magnitude

v(r̃, t̃) = ε3

∑j=1

A j(εr̃,ε2t̃)︸ ︷︷ ︸

∝exp(iεq j ·r̃)

eiq̃ j ·r f (z)+ c.c.+h.o.t.

∂tA1 = µA1 +(n1 ·∇)2A1 +A∗2A∗3 −A1|A1|

2

−(ν+ γ)A1|A2|2 − (ν− γ)A1|A3|2 − iβA1(τ̂1 ·∇)Q

+i(α1 −α3)A∗3 (n2 ·∇)A∗2 + i(α1 +α3)A

∗2 (n3 ·∇)A

∗3

+iα2 (A∗3 (τ2 ·∇)A∗2 −A

∗2 (τ3 ·∇)A

∗3)

∇2Q =3

∑j=1

[2(n̂ j ·∇)(τ̂ j ·∇)+ τ

((n̂ j ·∇)2 − (τ̂ j ·∇)2

)]|A j|

2

with drift velocity V ∝ ∇× (Qêz)

• Strong resonance A∗2A∗3 • Nonlinear gradient terms

Ginzburg-Landau Equations for Hexagons

~

~

~ τ1τ2

n2

n1

n3 τ3

q

q2

3

1q

• Envelope equations:Slow variation of wavevector and magnitude

v(r̃, t̃) = ε3

∑j=1

A j(εr̃,ε2t̃)︸ ︷︷ ︸

∝exp(iεq j ·r̃)

eiq̃ j ·r f (z)+ c.c.+h.o.t.

∂tA1 = µA1 +(n1 ·∇)2A1 +A∗2A∗3 −A1|A1|

2

−(ν+ γ)A1|A2|2 − (ν− γ)A1|A3|2 − iβA1(τ̂1 ·∇)Q

+i(α1 −α3)A∗3 (n2 ·∇)A∗2 + i(α1 +α3)A

∗2 (n3 ·∇)A

∗3

+iα2 (A∗3 (τ2 ·∇)A∗2 −A

∗2 (τ3 ·∇)A

∗3)

∇2Q =3

∑j=1

[2(n̂ j ·∇)(τ̂ j ·∇)+ τ

((n̂ j ·∇)2 − (τ̂ j ·∇)2

)]|A j|

2

with drift velocity V ∝ ∇× (Qêz)

• Strong resonance A∗2A∗3 • Rotation • Mean Flow

Ginzburg-Landau Equations for Hexagons

~

~

~ τ1τ2

n2

n1

n3 τ3

q

q2

3

1q

• Envelope equations:Slow variation of wavevector and magnitude

v(r̃, t̃) = ε3

∑j=1

A j(εr̃,ε2t̃)︸ ︷︷ ︸

∝exp(iεq j ·r̃)

eiq̃ j ·r f (z)+ c.c.+h.o.t.

∂tA1 = µA1 +(n1 ·∇)2A1 +A∗2A∗3 −A1|A1|

2

−(ν+ γ)A1|A2|2 − (ν− γ)A1|A3|2 − iβA1(τ̂1 ·∇)Q

+i(α1 −α3)A∗3 (n2 ·∇)A∗2 + i(α1 +α3)A

∗2 (n3 ·∇)A

∗3

+iα2 (A∗3 (τ2 ·∇)A∗2 −A

∗2 (τ3 ·∇)A

∗3)

∇2Q =3

∑j=1

[2(n̂ j ·∇)(τ̂ j ·∇)+ τ

((n̂ j ·∇)2 − (τ̂ j ·∇)2

)]|A j|

2

with drift velocity V ∝ ∇× (Qêz)

• Strong resonance A∗2A∗3 • Rotation • Mean flow

Penta-Hepta Defects

(Eckert and Thess, 1999)

• Defects in individual roll sys-tems: dislocations

• Resonant term A∗j−1A∗j+1:

strong attraction betweentwo dislocations indifferent roll systems⇒ penta-hepta defect

Side-bandInstabilities

InducedNucleation

-0.5 0 0.5Reduced Wave Number q

0

1

2

3

4

5

Con

trol

Par

amet

er µ osc. long-wave

steady long-wavesteady short-wave

~

~

~ τ1τ2

n2

n1

n3 τ3

q

q2

3

1q

v = ε3

∑j=1

A j eiq̃j·r

A1 . A2 A3

A1. A2 A3

• Temporal evolution of mean wavevector

0 500 1000 1500 2000 2500Time

0

1

2

3

Loc

al w

avev

ecto

r

β = 0

Transverse

Longitudinal

0 500 1000 1500 2000 2500Time

0

1

2

3

Loc

al w

aven

umbe

r

β = -0.1

Transv

erse

Longitudinal

- transient induced nucleation - ‘persistent’ chaos(Colinet, Nepomnyashchy, and Legros, 2002)

~

~

~ τ1τ2

n2

n1

n3 τ3

q

q2

3

1q

• Dependence on chiral, nonlinear gradient term α3

Separation of dislocations in PHD

0 0.1 0.2 0.3 0.4 0.5Chiral gradient term α3

0

0.2

0.4

0.6

0.8

1

Dis

tanc

e

β = 0

Limit of persistent creation of PHDs

0 0.2 0.4 0.6 0.8Chiral gradient term α3

-1.5

-1

-0.5

0M

ean

flow

βPHD stable

PHD unstable

Chaos

µ=1

Swift-Hohenberg Model

• Rotation and mean flow

∂tψ = Rψ− (∇2 +1)2ψ−ψ3 +α(∇ψ)2 +

γ êz · {∇ψ×∇4ψ}−U ·∇ψ

∇2ξ = êz · {∇ψ×∇4ψ}+δ{(4ψ)2 +∇ψ ·∇4ψ

}U = β(∂yξ,−∂xξ)

R = 0.17, Lx = 233, α = 0.4, β = −2.6, γ = 2, δ = 0.0067

Defect Proliferation

.

.

Defect Statistics in Stripe-Based Defect Chaos

• Complex Ginzburg-Landau equation(Gil, Lega, and Meunier, 1990)

• Electroconvection(Rehberg et al., 1989)

• Convection in inclined layer(Daniels and Bodenschatz, 2002)

(Gil et al., 1990)

• Statistical model for defect pairs(Gil et al., 1990)

– random pairwise creation– uncorrelated diffusion– pairwise annihilation upon collision

Squared Poisson distributionfor number of defects

P(n;ρ) ∝ρn

(n!)2

Penta-Hepta Defect Statistics

0 10 20 30 40Number of Dislocation Pairs

0

0.05

0.1

0.15

Rel

ativ

e F

requ

ency

L=233 (+,-,0)(+,0,-)(0,+,-)PPoisson Fit

Induced Nucleation

α

β

γ δ

P(n+12,n−12,n

+23,n

−23,n

+31,n

−31) → P(n

+12) ≡ P(n)

P(n+1)Γ−(n+1) = P(n)Γ+(n)

Γ−(n) = a1 n+a2 n2

Γ+(n) = c0 + c1 n+ c2 n2

For L = 244

a1 = α = 8.6

a2 = 2β+δ ≡ 1

c0 = γ = 20.07

c1 = 2α = 20.7

c2 = β = 0.12

Creation and Annihilation Rates

• follow defect trajectories:creation and annihilationrates

• consistent with distributionfunction

• spontaneous creation:locally unstable at low q

0 5 10 15 20Number of Dislocation Pairs

0

0.1

0.2

0.3

0.4

0.5

Rat

e

CreationAnnihilation

• Defect distribution function

0 5 10 15 20Number of Dislocation Pairs

0

0.05

0.1

0.15

0.2

0.25

Rel

ativ

e F

requ

ency

L=114

Poisson FitPHDP(n)

• Wavenumber distribution function

0.7 0.8 0.9 1 1.1Wavenumber q

0

0.01

0.02

0.03

0.04R

elat

ive

Fre

quen

cy

unstable stable

Summary:

Penta-Hepta Defect Chaos in Hexagons

• Induced Nucleation of Defects

• Proliferation of Defects

• Broad Defect Distribution Functionnot squared Poisson distribution

• Creation and Annihilation Ratesconsistent with distribution function

Parametrically Excited Standing Waves

• Coupled Ginzburg-Landau equations

∂tA+ s∂xA = dx∂2xA+dy∂2yA+aA+bB+

c|A|2A+g|B|2A

∂tB− s∂xB = d∗x ∂2xB+d

∗y ∂

2yB+a

∗B+bA+

c∗|B|2B+g∗|A|2B

Forcing f (t) ∼ b

h(x,t)f(t)

(cf. Faraday waves)

• Typical phase diagram: periodically forced Hopf bifurcation

0Linear growth rate ar

0

For

cing

Am

plit

ude

b

standing

waves

nowaves

travelingwaves

paritybreaking

• Patterns

OrderedChaos

L = 272

b = 0.7

DisorderedChaos

L = 272

b = 0.625

• Correlation Functions

0 10 20 30 40 50 60y-Location

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e Defect Loopsin

Space-Time

• Defect motionclimb ⇒ local change in wavelengthglide ⇒ local rotation of pattern

y

x

Statistics of Space-Time Loops

100

101

102

103

Number n of Defects in Loop

10-6

10-5

10-4

10-3

10-2

10-1

100

Rel

ativ

e F

requ

ency

n-1.5

b=0.4b=0.5 b=0.625 b=0.7b=1.0

∆

∆t

x

n=6

• ordered: exponential decay

• disordered: power-law decay

100

101

102

103

x-span ∆x10

-6

10-5

10-4

10-3

10-2

10-1

100

Rel

ativ

e F

requ

ency

∆x-3

b=0.4b=0.5b=0.625b=0.7b=1.0

101

102

103

t-span ∆t10

-6

10-5

10-4

10-3

10-2

10-1

100

Rel

ativ

e F

requ

ency

∆t-2.7

b=0.4b=0.5b=0.625b=0.7b=1.0

Loop Statistics in a Lattice Model

• Random creation of defect pairs

• Defects diffuse independentlyon lattice

100

101

102

103

104

10510

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Rel

ativ

e F

requ

ency

n-1.6

∆x-2.9∆t-2.4

n∆x∆y∆t

p=0.000016 & p=0.0001 L=1600

Loop Statistics in a Lattice Model

• Random creation of defect pairs

• Defects diffuse independentlyon lattice

α β γ

Coupled CGL 1.5 3 2.7

Lattice 1.6 2.9 2.4

100

101

102

103

104

10510

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Rel

ativ

e F

requ

ency

n-1.6

∆x-2.9∆t-2.4

n∆x∆y∆t

p=0.000016 & p=0.0001 L=1600

Single Complex Ginzburg-Landau Equation

• Generic description of Hopf bifurcation

∂tA = A+(1+ ib1)∇2A− (b3 − i)A|A|2

• Defect chaos above line T :creation and annihilation of defects

(Chaté & Manneville, 1996) 0 0.5 1 1.5 2 2.5b3

-2

0

2

4

b1

DefectPhase

Frozen

Vortices

L

BF

T

Stable

Plane Waves

Chaos

S2

Chaos

• Defect loop statistics

1 10 100 1000Number n of Defects in Loop

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

Cum

ul. R

elat

ive

Fre

quen

cy

n-1.5

b1=4 b3=0.5b1=2 b3=1b1=1 b3=0.5

1 10 100 1000x-span ∆x

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

Cum

ul. R

elat

ive

Fre

quen

cy

∆x-3

b1=4 b3=0.5b1=2 b3=1b1=1 b3=0.5

1 10 100 1000t-span ∆t

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

Cum

ul. R

elat

ive

Fre

quen

cy

∆t-2.5

b1=4 b3=0.5b1=2 b3=1b1=1 b3=0.5

Defects and Spatio-Temporal Chaos

• Penta-Hepta Defect Chaosinduced nucleation, proliferationbroad distribution(Y.-N. Young & HR, Physica D, to appear)

• Ordered ⇔ Disordered Spatio-Temporal Chaosdefect loops in space-timeexponential vs. algebraicdefect unbinding transition(G.D. Granzow & HR, Phys. Rev. Lett. 87 (2001) 174502)

0 10 20 30 40 50 60y-Location

15000

15200

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e

• Power Laws in Disordered Regime: Exponents

– Parametrically driven waves

– Lattice model

– Complex Ginzburg-Landau equation

www.esam.northwestern.edu/riecke

Parameters

Ginzburg-Landau:ν = 2, α1 = α2 = 0, α3 = 0.7, γ = 0.2, τ = 0.5.β = 0 and β = −0.1 as indicated

large SH:α = 0.4, γ = 2, β = −2.6, δ = 0??????, R = 0.17, and L = 233.

small SH:L = 114, α = 0.4, γ = 3, β = −5, δ = 0?????????????????? and R = 0.09.

Distribution function in SH: L = 244fit with a1 = 8.6, a2 = 1 (normalization), c0 = 20.07, c1 = 20.7, c2 = 0.12

update (1/15/2003): the creation rates are actually: L=114 c0=31.8 c1=3.3c2=.4 a1=5

Coupled Ginzburg-Landau equations:a = +0.25, c = −1+4i, d = 1+0.5i, s = 0.2, g = −1−12ib as indicated

Lattice Model:L = 1600, p = 0.000016

Farben fuer TEX (aus /usr/local/lib/tex/macros/dvips/colordvi.tex

GreenYellow Yellow Goldenrod Dandelion Apricot Peach Melon YellowOrangeOrange BurntOrange Bittersweet RedOrange Mahogany Maroon BrickRedRed OrangeRed RubineRed WildStrawberry Salmon CarnationPinkMagenta VioletRed Rhodamine Mulberry RedViolet Fuchsia Lavender ThistleOrchid DarkOrchid Purple Plum Violet RoyalPurple BlueVioletPeriwinkleCadetBlue CornflowerBlue MidnightBlue NavyBlue RoyalBlueBlue Cerulean Cyan ProcessBlue SkyBlue Turquoise TealBlue AquamarineBlueGreen Emerald JungleGreen SeaGreen Green ForestGreenPineGreen LimeGreen YellowGreen SpringGreen OliveGreen RawSiennaSepia Brown Tan Gray Black White

aαbβaαbβaαbβ

a α

acroread erst maximieren bevor auf full-screen gegangen wird

check ps2pdf: what’s differentbetter picture of undulation chaoswarum bekommt α2 Term keinen Rotationsterm? warum faktor 2

in Gamma+-

Notes:

Kl Faraday mean flow not to be expected to be knownmention only when actually used

PSW: don’t discuss Faraday as the relevant aspect: this is a model thatshows a certain transition and it happens to be relevant to Faraday fornegative growth rate

spatially extended dynamical systems − > structured patterened states

Greenside: RB, excitation waves in heart

disordered in space and chaotic in time

exhibit defects: striking features of pattern

use for characterization and description of spatio-temporal chaos

present results of 2 lines of research

hexagons YY

defect trajectories