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Complex Patterns:
Diagnostics and Excitation
with
Santiago MadrugaMPI Complex Systems, Germany
Werner PeschU. Bayreuth, Germany
Jessica ConwayNorthwestern University
supported by DOE and NSF
Spatio-Temporal Chaos
Undulation Chaos
(Daniels and Bodenschatz, 2002)
T+dT
T
up
down
Kuppers-Lortz Chaos
Rotation rate Ω = 12.1, ǫ = 0.2
(Hu, Ecke, and Ahlers, 1997)
Spatio-Temporal Chaos
Undulation Chaos
(Daniels and Bodenschatz, 2002)
T+dT
T
up
down
Kuppers-Lortz Chaos
Rotation rate Ω = 19.8, ǫ = 0.18
(Hu, Ecke, and Ahlers, 1997)
Spiral Defect Chaos
Convection at low Prandtl numbers
(Morris, Bodenschatz, Cannell, Ahlers, 1993)
Pr = 1.5
Boussinesq
Pr = 0.3
Non-Boussinesq
Defect Chaos in Belousov-Zhabotinsky Reaction
• Transition from spiral waves to defect chaos
(Ouyang and Flesselles, 1996)
Periodically Forced Belousov-Zhabotinsky Reaction
Near-resonant forcing of chemical oscillations
(Lin et al., 2004)
⇐ 1:2 Labyrinth
3:2 Cellular ⇒
(Petrov, Ouyang, & Swinney, 1997)
(Lin et al., 2004)
1:3 Spirals
⇐ weaker forcing
stronger forcing ⇒
(Lin et al., 2004)
Characterization of Complex Patterns
• Correlation functions, spectral entropy
• Local wavevector (Egolf, Melnikov, and Bodenschatz, 1998)
• Disorder function (Gunaratne, Hoffman, and Kouri, 1998)
• Statistics for number of dislocations
- complex Ginzburg-Landau equation (Gil, Lega, and Meunier, 1990)
- electroconvection (Rehberg et al., 1989)
- reaction-diffusion system (Hildebrand, Bär, and Eiswirth, 1995)
- undulation chaos (Daniels and Bodenschatz, 2002)
- penta-hepta defect chaos (Young and HR, 2003)
• Statistics of defect trajectories
- ‘unbinding’ transition of defect pairs (Granzow and HR, 2001)
• Homology: components and holes (Krishan et al., 2007)
Geometric Diagnostics of Spiral Defect Chaos
• automated analysis of experimentally accessible data
• mid-level contour
• smoothing of
contours
• segments:
tip → vertex
• number of closed contours
(cf. Schatz, Mischaikow; Krishan et al., 2007)
• contour length
• compactness of contour
• arclength of segments
• winding number of segments
• statistics based on
1000-2000 snapshots
Number of Components
Pr = 0.3 Pr = 1.5Topological measure
(Betti numbers)
(cf. Krishan et al., 2007)
0.5 1 1.5Red. Rayleigh Number ε
0
10
20
30
40
50
# C
lose
d C
onto
urs
Pr=1.5Pr=1.0Pr=0.3
Number of closed contours
• increases with increasing ǫ
• decreases with increasing Pr
Mean Arclength
Pr = 0.3 Pr = 1.0 Pr = 1.5
ǫ = 1
0.5 1 1.5Red. Rayleigh Number ε
0
100
200
300
Arc
leng
th
Pr=1.5Pr=1.0Pr=0.3
• arclength decreases
with increasing ǫ
• arclength increases
with increasing Pr
Standard Deviation of Winding Number
Pr = 0.3 Pr = 1.0 Pr = 1.5
ǫ = 1
0.5 1 1.5Red. Rayleigh Number ε
0
0.2
0.4
0.6
0.8
1
SD
Win
ding
Num
ber
Pr=1.5Pr=1.0Pr=0.3
• winding number decreases
with increasing ǫ
• winding number increases
with increasing Pr
• dependence not only.
on ǫ − ǫSDC
Compactness: Targets and Bubbles
Pr = 0.3 Pr = 1.5Compactness:
C = 4πA
P2
A = area
P = perimeter
Compactness: Targets and Bubbles
Pr = 0.3 Pr = 1.5Compactness:
C = 4πA
P2
circular: C ∼ 1
elongated:
C ∼ 4π λP/2P2 ∝ P−1
−6−5
−4−3
−2−1
012345678910
0
0.01
0.02
0.03
0.04
Ln (Compactness)Ln (Contourlength)
−6−5
−4−3
−2−1
012345678910
0
0.02
0.04
0.06
Ln (Compactness)Ln (Contourlength)
Winding Number and Arclength of Contours
• Large Pr:
large segments are
spirals
• Small Pr:
few clear spirals
• Winding number has
exponential distribution
(cf. Hu & Ecke, 1997)
0 1 2 3 4 5|Winding Number|
0.0001
0.001
0.01
0.1
1R
el. F
requ
ency
ε=0.7ε=1.0ε=1.4
Winding Number and Arclength of Contours
• Large Pr:
large segments are
spirals
• Small Pr:
few clear spirals
• Winding number has
exponential distribution
(cf. Hu & Ecke, 1997)
• Decay rate depends
on Pr
0 2 4 6 8|Winding Number|
0.0001
0.001
0.01
0.1
1R
el. F
requ
ency
Pr=1.5Pr=1.0Pr=0.3
Spiral Statistics
Distribution function for spiral number
• for ǫ = 1 and Pr = 1 narrower than Poisson distribution
(cf. Ecke et al., 1995; Ecke & Hu, 1997)
• collapse for different spiral thresholds
0 10 20 30 40 50Number of Spirals
0
0.05
0.1
0.15
0.2
Rel
. Fre
quen
cy
quarter-turn spiralshalf-turn spiralsfull-turn spiralsPoisson
-3 -2 -1 0 1 2 3
(# Spirals - Mean) /(Mean)1/2
0
0.2
0.4
0.6
Rel
. Fre
quen
cy
quarter turnhalf turnfull turn
Dependence on Vertical Position
So far all measurements were at the mid-plane z = 0
• Distribution function for
the winding number
0 1 2|Winding Number|
0.001
0.01
0.1
Rel
. Fre
quen
cy
z=-0.125z=-0.25z=0
• Mean quantities
-0.2 -0.1 0 0.1 0.2 0.3
Vertical Position z
0.6
0.8
1
1.2
1.4
Enh
ance
men
t ove
r z=
0
# `White’ Contours# `Black’ Contours# All Contours# SpiralsArc LengthSD Winding Number
Non-Boussinesq Effects
Temperature-dependent
fluid properties
• break up-down symmetry
• introduce
quadratic triad interaction
CO2: h = 0.080cm, T0 = 40C
3 4 5 6Wave number q
0
0.2
0.4
0.6
0.8
1
Red
uced
Ray
leig
h N
umbe
r ε
Stab
le
Am
plitu
de u
nsta
ble
Non-Boussinesq Effects
Temperature-dependent
fluid properties
• break up-down symmetry
• introduce
quadratic triad interaction
CO2: h = 0.052cm, T0 = 27C
2.4 2.6 2.8 3 3.2Wave number q
0
0.2
0.4
0.6
0.8
1
Red
uced
Ray
leig
h N
umbe
r ε
Stable hexagons
• Longitudinal
phase mode
ǫ = 0.195, q = 2.5
• Transverse
phase mode
ǫ = 0.16, q = 3.1
Non-Boussinesq Effects
Temperature-dependent
fluid properties
• break up-down symmetry
• introduce
quadratic triad interaction
CO2: h = 0.052cm, T0 = 27C
2.4 2.6 2.8 3 3.2Wave number q
0
0.2
0.4
0.6
0.8
1
Red
uced
Ray
leig
h N
umbe
r ε
Stable hexagons
• Transient
rolls
ǫ = 0.3
q = 3.12
Non-Boussinesq Spiral Defect Chaos
• Working fluid SF6
h = 0.0542cm
p = 140psi
T0 = 80C
Boussinesq Non-Boussinesq
• Number of components
strongly enhanced
0 0.2 0.4 0.6 0.8 1 1.2 1.4Reduced Rayleigh Number ε
0
10
20
30
40N
o. o
f Clo
sed
Con
tour
sNBOB
Non-Boussinesq Spiral Defect Chaos
• Working fluid SF6
h = 0.0542cm
p = 140psi
T0 = 80C
Boussinesq Non-Boussinesq
• Many small components
−4−3
−2−1
012345678910
0
0.01
0.02
0.03
0.04
0.05
0.06
Ln(Contourlength)Ln(Compactness)
(a)
Rel
ativ
e F
requ
ency
−4−3
−2−1
012345678910
0
0.02
0.04
0.06
Ln(Compactness)Ln(Contourlength)
(b)
Rel
ativ
e F
requ
ency
Complex Patterns in the Faraday System
(Kudrolli, Pier, & Gollub, 1992)
• Control of mode interaction by
choice of forcing function
(Arbell & Fineberg, 2002)
Complex Patterns in Resonantly ForcedComplex Ginzburg-Landau Equation
Weakly nonlinear oscillations forced at multiple resonances
∂tA = (1 + iβ)∇2A + (µ + iσ)A − (1 + iα)|A|2A
+(
cos χ + sinχeiνt)
γA∗ + ηA∗2
• 1:2 resonance: spatial
patterns through dispersion
• 1:3 resonance: quadratic
resonant triad interaction
• time-dependent coefficient:
subharmonic patterns ⇒
no transcritical hexagons
• Floquet ansatz (µ < 0)
0 0.5 1 1.5 2 2.5 3Wave Number k
5
10
15
20
25
1:2
Forc
ing
Stre
ngth
γ0.4 0.6 0.8 1 1.2
k
2.4
2.5
2.6
γ
Pattern Selection through Weakly Damped Modes
Weakly nonlinear theory (µ < 0)
A = ǫ(
A1eik1·r + A2e
ik2·r + . . .)
F (t) + ǫ2B(r, t) + O(ǫ3)
• Quadratic order:
strong excitation of
weakly damped
harmonic modes
• Cubic order:
feed-back into
amplitude equation
for fundamentals
dA1
dt= λA1 − b0A1|A1|
2 −
−b(θ)|A1A2|2 + . . .
k1
θr
k2+k 21k
2k2
Pattern Selection through Weakly Damped Modes
Weakly nonlinear theory (µ < 0)
A = ǫ(
A1eik1·r + A2e
ik2·r + . . .)
F (t) + ǫ2B(r, t) + O(ǫ3)
η = ρ eiπ/4 k(H)c = 2 k
(SH)c
0 20 40 60 80Rhomb Angle θ
0
0.5
1
1.5
2
Cro
ss C
oupl
ing
b(θ)
/b0
ρ=1ρ=1.5
dA1
dt= λA1 − b0A1|A1|
2 −
−b(θ)|A1A2|2 + . . .
k1
θr
k2+k 21k
2k2
(Viñals et al.)
Pattern Selection through Weakly Damped Modes
Weakly nonlinear theory (µ < 0)
A = ǫ(
A1eik1·r + A2e
ik2·r + . . .)
F (t) + ǫ2B(r, t) + O(ǫ3)
|η| = 1 k(H)c = 1.86 k
(sh)c
0 20 40 60 80Rhomb Angle θ
-30
-20
-10
0
Cro
ss C
oupl
ing
b(θ)
/b0
η=eπi/4
η=e3πi/4
dA1
dt= λA1 − b0A1|A1|
2 −
−b(θ)|A1A2|2 + . . .
k1
+k 21k
θr
k2
22k
(Silber et al.)
Stability of Super-Lattice and Quasi-Patterns
• Cross-coupling coefficient b(θ)/b0: stability of multi-mode patterns
• Conditions for relative stability of patterns on the same Fourier grid
• Overview of ‘preferred’ patterns: Lyapunov functional
η = ρ eiπ/4 k(H)c = 2 k
(SH)c
0 0.5 1 1.5 21:3 Forcing Strength ρ
-0.08
-0.06
-0.04
-0.02
0
Lyap
unov
Ene
rgy
StripesSquaresHexagons4-mode Quasipattern5-mode Quasipattern6-mode Quasipattern
0.6 0.8 1 1.2ρ
-0.05
-0.04
-0.03
Numerical Simulations
Starting from random initial conditions
η = 0.8 eiπ/4
Numerical Simulations
Starting from random initial conditions
η = 0.9 eiπ/4
Numerical Simulations
Starting from random initial conditions
η = 1.2 eiπ/4
Numerical Simulations
Starting from random initial conditions
η = 2 eiπ/4
Numerical Simulations
Starting from random initial conditions
η = 3 eiπ/4
Numerical Simulations
Starting from random initial conditions
Spectral Entropy
5000 10000 15000 20000Period T
5.5
6
6.5
7
7.5
8
8.5
Ent
ropy
S
ρ=0.8ρ=0.9ρ=1ρ=1.1ρ=1.2ρ=1.3ρ=1.4ρ=1.5ρ=1.8ρ=2ρ=3
Conclusions
Geometric Analysis of Patterns
• based on experimentally accessible data
• here applied to spiral defect chaos
• sub-Poisson distribution for number of spirals
• exponential distribution for winding number
• many small components for small Prandtl and non-Boussinesq
• patterns depend not only on distance from SDC onset
Quasi-Patterns in Multi-Resonance Complex Ginzburg-Land au Equation
• 3-mode lattices, 4-mode and 5-mode quasipatterns
• additional complexity from Hopf bifurcation?
www.esam.northwestern.edu/riecke