Spatially-extended dynamical systemsand pattern formation

Torino, January-March 2010

a.provenzale@isac.cnr.it

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

Pattern:

An organized, long-lived non-homogeneous stateof a physical, chemical

or biological system.

Physical patterns

Convection:S. Ciliberto et al., Phys. Rev. Lett. 61, 1198 (1988)

Physical patterns

Atmospheric convection. Photo by Hezi Yizhaq, Sede Boker, Negev desert

Patterns

Geomorphology: aeolian ripples. Wadi Rum desert

Geomorphology: aeolian ripples. Wadi Rum desert, Jordan

Erosion patterns in badlands (Tzin valley, Israel)

http://daac.gsfc.nasa.gov/ DAAC_DOCS/geomorphologyhttp://daac.gsfc.nasa.gov/ DAAC_DOCS/geomorphology

River networks(Yemen)

Chemical and biological patterns

A.T. Winfree et al,  Phisica D8, 35 (1983)

Biological patterns

S. Kondo et al, Nature 376, 765 (1995)

J.D. Murray, J. Theor. Biol. 88,

161 (1981)

Vegetation patterns at landscape scale

Rietkerk et al., The American Naturalist 160 (4), 2002

50 m

Vegetation patterns at landscape scale

Valentin et al., Catena 37, 1-24 (1999)

Coherent structures: vortices

Coherent structures: vortices

Coherent structures: vortices

Coherent structures: vortices

Coherent structures: convective plumes

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

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)

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( )

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( )

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( )

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( )

Characteristics

€

x = z + c0t

t

x

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

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

This is a linear equation (superposition)

This is a non-dispersive equation

This is a conservative equation

€

∂u

∂t+ c0

∂u

∂ x= 0

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

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

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 !

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

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