-
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
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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
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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
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Non-Boussinesq Convection
(Bodenschatz, de Bruyn, Ahlers, Cannell, 1991)
T+dT
T
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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
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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
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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
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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
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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
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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
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• 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
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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
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Defect Proliferation
.
.
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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
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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
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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
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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
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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
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• Patterns
OrderedChaos
L = 272
b = 0.7
DisorderedChaos
L = 272
b = 0.625
• Correlation Functions
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0 10 20 30 40 50 60y-Location
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15600T
ime
0 10 20 30 40 50 60y-Location
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Tim
e Defect Loopsin
Space-Time
• Defect motionclimb ⇒ local change in wavelengthglide ⇒ local rotation of pattern
y
x
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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
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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
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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
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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
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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
15400
15600
Tim
e
• Power Laws in Disordered Regime: Exponents
– Parametrically driven waves
– Lattice model
– Complex Ginzburg-Landau equation
www.esam.northwestern.edu/riecke
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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
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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
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in Gamma+-
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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
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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