spatially-extended dynamical systems and pattern formation torino, january-march 2010
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Spatially-extended dynamical systems and pattern formation Torino, January-March 2010 [email protected]. Goal: provide an introduction to the behavior of spatially-extended dynamical systems and to the study of PDEs. an introduction to pattern formation - PowerPoint PPT PresentationTRANSCRIPT
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Spatially-extended dynamical systemsand pattern formation
Torino, January-March 2010
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Goal:
provide an introduction to the behaviorof spatially-extended dynamical systems
and to the study of PDEs.
an introduction to pattern formationand the study of nonlinear PDEs
applications: geosciences, biology, ecology
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Pattern:
An organized, long-lived non-homogeneous stateof a physical, chemical
or biological system.
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Physical patterns
Convection:S. Ciliberto et al., Phys. Rev. Lett. 61, 1198 (1988)
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Physical patterns
Atmospheric convection. Photo by Hezi Yizhaq, Sede Boker, Negev desert
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Patterns
Geomorphology: aeolian ripples. Wadi Rum desert
Geomorphology: aeolian ripples. Wadi Rum desert, Jordan
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Erosion patterns in badlands (Tzin valley, Israel)
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http://daac.gsfc.nasa.gov/ DAAC_DOCS/geomorphologyhttp://daac.gsfc.nasa.gov/ DAAC_DOCS/geomorphology
River networks(Yemen)
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Chemical and biological patterns
A.T. Winfree et al, Phisica D8, 35 (1983)
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Biological patterns
S. Kondo et al, Nature 376, 765 (1995)
J.D. Murray, J. Theor. Biol. 88,
161 (1981)
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Vegetation patterns at landscape scale
Rietkerk et al., The American Naturalist 160 (4), 2002
50 m
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Vegetation patterns at landscape scale
Valentin et al., Catena 37, 1-24 (1999)
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Coherent structures: vortices
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Coherent structures: vortices
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Coherent structures: vortices
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Coherent structures: vortices
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Coherent structures: convective plumes
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Contents:
1. Intro to extended systems and PDE: heat eq., Fisher eq.
2. Navier-Stokes eqns. and fluid dynamics
3. Convection: linear stability, nonlinear saturation, patterns
4. Turbulence and transport
5. The Atmospheric Planetary Boundary Layer
6. Rotating fluids, waves and vortices
7. Turing mechanism and chemical patterns
8. Patterns in geomorphology: aeolian ripples
9. Vegetation patterns
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Introduction to spatially-extended dynamical systems
Types of mathematical descriptions:
PDEs (t=c, x=c, f=c)Coupled ODEs (t=c, x=d, f=c)Coupled maps (t=d, x=d, f=c)
Cellular automata (t=d, x=d, f=d)
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PDE for time evolution:initial value problem
€
∂ ru
∂t=
r K
r u , ∂
r u ,....[ ] ;
r u =
r u
r x , t( )
r u
r x , t = 0( ) =
r u 0
r x ( )
r u S, t( ) =
r u S S( )
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PDE for boundary value problems
€
0 =r K 0
r u , ∂
r u ,....[ ] ;
r u =
r u
r x , t( )
r u S, t( ) =
r u S S, t( )
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PDE for time evolution:look for stationary solutions,
homogeneous solutions, special solutions (eg travelling waves),
general solutions
€
∂ ru
∂t= 0 ; ∇ ui = 0
r u =
r u x − c t( )
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The simplest PDE
€
∂u
∂t+ c0
∂u
∂ x= 0 ; u = u x, t( )
z = x − c0t
∂
∂t=
∂z
∂t
∂
∂z= −c0
∂
∂z,
∂
∂ x=
∂z
∂ x
∂
∂z=
∂
∂z
−c0
∂u
∂z+ c0
∂u
∂z= 0
u = u(z) = u x − c0t( )
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Characteristics
€
x = z + c0t
t
x
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Another approach: Fourier decomposition
€
u = u x, t( ) x →±∞ ⏐ → ⏐ ⏐ 0
u(x, t) =1
2πˆ u (k, t)
−∞
∞
∫ exp ikx( )dk , ˆ u (k, t) = u(x, t)−∞
∞
∫ exp −ikx( ) dx
ˆ u (k, t) = ˆ u (k)A(t) = ˆ u (k)exp −iω(k)t[ ]
u(x, t) =1
2πˆ u (k)
−∞
∞
∫ exp ikx − iω(k)t[ ]dk , ˆ u (k) = u(x,0)−∞
∞
∫ exp −ikx( )dx
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Another approach: Fourier decomposition
€
∂u
∂t+ c0
∂u
∂ x= 0 ; u = u x, t( ) x →±∞
⏐ → ⏐ ⏐ 0
∂
∂t
1
2πˆ u (k)
−∞
∞
∫ exp ikx − iω(k)t[ ]dk ⎧ ⎨ ⎩
⎫ ⎬ ⎭+ c0
∂
∂ x
1
2πˆ u (k)
−∞
∞
∫ exp ikx − iω(k)t[ ]dk ⎧ ⎨ ⎩
⎫ ⎬ ⎭= 0
−iω(k)+ ic0 k = 0 ⇒ ω(k) = c0 k
u(x, t) = ˆ u (k)−∞
∞
∫ exp ik x − c0 t( )[ ]dk
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This is a linear equation (superposition)
This is a non-dispersive equation
This is a conservative equation
€
∂u
∂t+ c0
∂u
∂ x= 0
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This is a conservative equation(also, the amplitude of each Fourier component)€
∂u
∂t+ c0
∂u
∂ x= 0
U(t) = u(x, t)dx−∞
∞
∫ ; E(t) = u2(x, t)dx−∞
∞
∫
dU
dt= 0 ;
dE
dt= 0
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A slightly more complicated equation
€
∂u
∂t+ c0
∂u
∂ x+ β
∂ 3 u
∂ x 3= 0
ω(k) = c0k − β k 3
u =1
2πˆ u (k)
−∞
∞
∫ exp i kx − c0k − β k 3( ) t[ ]{ }dk
Still linear and conservative, but now dispersive
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A different case
€
∂u
∂t+ c0
∂u
∂ x− μ
∂ 2 u
∂ x 2= 0
iω(k) = ic0k + μ k 2
u =1
2πˆ u (k)
−∞
∞
∫ exp i kx − c0k t[ ] − μ k 2 t{ }dk
u =1
2πˆ u (k)
−∞
∞
∫ exp i kx − c0k t[ ]{ }exp −μ k 2 t[ ]dk
Still linear but now dissipative !
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Homogeneous state and its stability
€
∂u
∂t+ c0
∂u
∂ x− μ
∂ 2 u
∂ x 2= 0
u = 0stable and attracting if μ > 0
unstable if μ < 0
⎧ ⎨ ⎩
for μ = 0 there is a bifurcation
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Finally, our most complicated linear PDE…
€
∂u
∂t+ c0
∂u
∂ x+ β
∂ 3 u
∂ x 3− μ
∂ 2 u
∂ x 2= 0
iω(k) = i c0k − β k 3[ ] + μ k 2
u =1
2πˆ u (k)
−∞
∞
∫ exp i kx − c0k − β k 3( ) t[ ] − μ k 2 t{ }dk
u =1
2πˆ u (k)
−∞
∞
∫ exp i kx − c0k − β k 3( ) t[ ]{ }exp −μ k 2 t[ ]dk
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