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Cours : Dynamique Non-Linéaire
Laurette [email protected]
VII. Reaction-Diffusion Equations:
1. Excitability
2. Turing patterns
3. Lyapunov functionals
4. Spatial analysis and fronts
Reaction-Diffusion Systems
∂tui = fi(u1, u2, . . .)︸ ︷︷ ︸
reaction
+ Di∆ui︸ ︷︷ ︸diffusion
Reactions fi couple different species ui at same location
Diffusivity Di couples same species ui at different locations
Describe oscillating chemical reactions, such as famous Belousov-Zhabotinskii
reaction, discovered by two Soviet scientists in 1950s-1960s.
Also describe phenomena in
–biology (population biology, epidemiology, neurosciences)
–social sciences (economics, demography)
–physics
Two species Spatially homogeneous
∂tu = f(u, v) + Du∆u ∂tu = f(u, v)∂tv = g(u, v) + Dv∆v ∂tv = g(u, v)
FitzHugh-Nagumo model Barkley model
f(u, v) = u− u3/3− v + I f(u, v) = 1ǫu(1− u)
(u− v+b
a
)
g(u, v) = 0.08 (u + 0.7− 0.8 v) g(u, v) = u− v
u-nullclines f(u, v) = 0 , v-nullclines g(u, v) = 0 , • steady statesstable if eigenvalues of
(fu fvgu gv
)
have negative real parts
Excitability
f(u, v) = 1ǫu(1− u)
(u− v+b
a
)g(u, v) = u− v
∂tu = f = 0 separates ←− and −→ O(ǫ−1)∂tv = g = 0 separates ↑ and ↓ O(1)
u = 1 excited phaseu = 0 v ∼ 1 refractory phaseu = 0 v ≪ 1 excitable phaseu = (v + b)/a excitation threshold
Waves in Excitable Medium
Spatial variation + diffusion + excitability =⇒ propagating waves
Excitable media in physiology:
–neurons
–cardiac tissue (the heart)
Pacemaker periodically emits electrical signals, propagated to rest of heart
Simulations from Barkley model, Scholarpedia
Spiral waves in 2D Spiral waves in 3D
Turing patterns
Instability of homogeneous solutions (ū, v̄) to reaction-diffusion systems{
0 = f(ū, v̄)0 = g(ū, v̄)
}
=⇒{
0 = f(ū, v̄) + Du∆ū0 = g(ū, v̄) + Dv∆v̄
}
What about stability? Does diffusion damp spatial variations?
Linear stability analysis:{
u(x, t) = ū + ũeσt+ik·x
v(x, t) = v̄ + ṽeσt+ik·x
}
=⇒{
σũ = fuũ + fvṽ −Duk2ũσṽ = guũ + gvṽ −Dvk2ṽ
}
Mk ≡(
fu −Duk2 fvgu gv −Dvk2
)
=
(fu fvgu gv
)
− k2(
Du 00 Dv
)
If Du = Dv ≡ D, thenσk± = σ0± − k2D ≤ σ0±
(ū, v̄) stable to homogeneous perturbations =⇒(ū, v̄) stable to inhomogeneous perturbations. Diffusion is stabilizing.
Alan Turing (famous WW II UK cryptologist, founder of computer science)
1952: homogeneous state can be unstable if Du 6= Dv
For instability, need Trk > 0 or Detk < 0
For instability, need Trk > 0 or Detk < 0
Homogeneous stability⇐⇒{
Tr0 = fu + gv < 0 andDet0 = fugv − fvgu > 0
}
Trk = fu + gv − (Du + Dv)k2 = Tr0 − (Du + Dv)k2 < Tr0 < 0So for instability, need Detk < 0
Detk = fugv − fvgu + DuDvk4 − (Dvfu + Dugv)k2= Det0︸ ︷︷ ︸
>0
+ DuDvk4
︸ ︷︷ ︸
>0, dominates for k≫1
−(Dvfu + Dugv)k2
Find negative minimum for intermediate k2:
0 =d Detk
dk2
∣∣∣∣k∗
= 2DuDvk2∗ − (Dvfu + Dugv)
k2∗ =Dvfu + Dugv
2DuDv=⇒ need Dvfu + Dugv > 0
Need Detk < 0 at k2∗ = (Dvfu + Dugv)/(2DuDv):
0 > Detk|k∗ = Det0 + DuDvk4∗ − (Dvfu + Dugv)k2∗
= Det0 +(Dvfu + Dugv)
2
4DuDv− 2(Dvfu + Dugv)
2
4DuDv
= Det0 −(Dvfu + Dugv)
2
4DuDv0 > 4DuDv(fugv − fvgu)− (Dvfu + Dugv)2
Collecting the four conditions:
Tr0 = fu + gv < 0
Det0 = fugv − fvgu > 02DuDvk
2∗ = Dvfu + Dugv > 0
4DuDv Detk|k∗ = 4DuDv(fugv − fvgu)− (Dvfu + Dugv)2 < 0
Turing patterns were first produced experimentally:
–in 1990 by de Kepper et al. at Univ. of Bordeaux
–in 1992 by Swinney et al. at Univ. of Texas at Austin
Turing pattern in a chlorite-
iodide-malonic acid chemical
laboratory experiment. From
R.D. Vigil, Q. Ouyang &
H.L. Swinney, Turing patterns in
a simple gel reactor, Physica A
188, 17 (1992)
Might be mechanism for:
–differentiation within embryos
–formation of patterns on animal coats, e.g. zebras and leopards
Lyapunov functionals
1D systems: no limit cycles, usually just convergence to fixed point
Generalize to multidimensional variational, potential, or gradient flows:
du
dt= −∇Φ ⇐⇒ dui
dt= −∂Φ
∂ui
For gradient flow, Jacobian is Hessian matrix:
H = −
∂2Φ/(∂u1∂u1) ∂2Φ/(∂u1∂u2) . . .
∂2Φ/(∂u2∂u1) ∂2Φ/(∂u2∂u2) . . .
......
...
H symmetric =⇒ no complex eigenvalues =⇒ no Hopf bifurcationsdΦ
dt=
∑
i
∂Φ
∂ui
dui
dt= −
∑
i
∂Φ
∂ui
∂Φ
∂ui= −|∇Φ|2
Φ decreases monotonically, either to−∞ or to point wheredu/dt = −∇Φ = 0 =⇒ no limit cycles
Generalize to reaction-diffusion systems involving potential Φ(u):
∂u
∂t= −∇Φ + ∂
2u
∂x2on xlo ≤ x ≤ xhi
Boundary conditions:
Dirichlet u(xlo) = ulo u(xhi) = uhi
or Neumann (homogeneous) ∂u∂x
(xlo) = 0∂u∂x
(xhi) = 0
Define free energy or Lyapunov functional:
F(u) ≡∫ xhi
xlo
dx
[
Φ(u(x, t))︸ ︷︷ ︸
potential energy
+1
2
∣∣∣∣
∂u(x, t))
∂x
∣∣∣∣
2
︸ ︷︷ ︸
kinetic energy
]
Seek quantity analogous to gradient:
F (x + dx) = F(x) +∇F(x) · dx + O(|dx|)2 for all dxThe functional derivative δF/δu is defined to be such that
F(u + δu) = F(u) +∫ xhi
xlo
dxδFδu· δu + O(δu)2 for every δu
Expand:
F(u + δu) =∫ xhi
xlo
dx
[
Φ(u + δu) +1
2
∣∣∣∣
∂(u + δu)
∂x
∣∣∣∣
2]
=
∫ xhi
xlo
dx
[
Φ(u) +∇Φ(u) · δu + . . . + 12
∣∣∣∣
∂u
∂x+
∂δu
∂x+ . . .
∣∣∣∣
2]
=
∫ xhi
xlo
dx
[
Φ(u) +1
2
∣∣∣∣
∂u
∂x
∣∣∣∣
2]
+
∫ xhi
xlo
dx
[
∇Φ(u) · δu + ∂u∂x· ∂δu∂x
]
+ O(δu)2
Integrate by parts:
∫ xhi
xlo
dx∂u
∂x· ∂δu∂x
=
[∂u
∂x· δu
]xhi
xlo
−∫ xhi
xlo
dx∂2u
∂x2· δu
Surface term vanishes since
{∂u∂x
(xlo) =∂u∂x
(xhi) = 0 for Neumann BCs
δu(xlo) = δu(xhi) = 0 for Dirichlet BCs
F(u+δu)=∫ xhi
xlo
dx
[
Φ(u) +
∣∣∣∣
∂u
∂x
∣∣∣∣
2]
+
∫ xhi
xlo
dx
[
∇Φ(u)− ∂2u
∂x2
]
· δu+O(δu)2
The functional derivative δF/δu is defined to be such that
F(u + δu) = F(u) +∫ xhi
xlo
dxδFδu· δu + O(δu)2 for every δu =⇒
∫ xhi
xlo
dxδFδu· δu =
∫ xhi
xlo
dx
[
∇Φ(u)− ∂2u
∂x2
]
· δu
Choosing δu to be delta function centered on any x and pointing in anyvector direction leads to pointwise equality:
δFδu
= ∇Φ(u)− ∂2u
∂x2= −∂u
∂t
dFdt
= lim∆t→0
1
∆t[F(t + δt)− F(t)]
= lim∆t→0
1
∆t[F(u(t + ∆t))−F(u(t))]
= lim∆t→0
1
∆t
[
F(
u(t) +∂u
∂t∆t + . . .
)
−F(u(t))]
= lim∆t→0
1
∆t
[
F(u(t)) +∫ xhi
xlo
dxδFδu· ∂u∂t
∆t + . . .−F(u(t))]
= lim∆t→0
1
∆t
[∫ xhi
xlo
dxδFδu· ∂u∂t
∆t + . . .
]
=
∫ xhi
xlo
dxδFδu· ∂u∂t
=
∫ xhi
xlo
dx
(
−∂u∂t
)
· ∂u∂t
= −∫ xhi
xlo
dx
∣∣∣∣
∂u
∂t
∣∣∣∣
2
≤ 0
F decreases so limit cycles cannot occur. Can be applied in higher spatialdimensions via volume integration and Gauss’s Divergence Theorem.
Spatial Analysis and Fronts
∂u
∂t= −dΦ
du+
∂2u
∂x2
Travelling wave solutions:
u(x, t) = U(x− ct) with c = 0 for steady statesξ ≡ x− ct
∂u
∂t(x, t) = −c dU
dξ(ξ)
∂2u
∂x2(x, t) =
d2U
dξ2(ξ)
Equation obeyed by steady states and travelling waves becomes
−c dudξ
= −dΦdu
+d2u
dξ2=⇒ d
2u
dξ2=
dΦ
du−c du
dξ
Analogy between space and time =⇒ x must be 1D
Spatial analysis or Mechanical analogy
d2u
dξ2︸︷︷︸
“acceleration”
= − d(−Φ)du︸ ︷︷ ︸
“potential gradient”
−c dudξ
︸ ︷︷ ︸
“friction”
u position ξ time
dudξ
velocity −Φ potential E(ξ) ≡ −Φ + 12
(dudξ
)2
energy
d2udξ2
acceleration −cdudξ
friction
Ė =dE
dξ=
d
dξ
[
−Φ + 12
(du
dξ
)2]
= −dΦdu
du
dξ+
du
dξ
d2u
dξ2
=
[
−dΦdu
+d2u
dξ2
]du
dξ= −c
(du
dξ
)2
< 0 if c > 0= 0 if c = 0> 0 if c < 0
c < 0⇐⇒{
“Increase in energy”
“Negative friction”
}
⇐⇒ just leftwards motion
If c = 0, then E constant with E = −Φ(u(ξ)) + 12
(du
dξ
)2
E + Φ(u(ξ)) =1
2
(du
dξ
)2
√
2(E + Φ(u(ξ))) =du
dξ∫
dξ =
∫du
√
2(E + Φ(u))
[ξ] ξξlo =
∫ u(ξ)
ulo
du√
2(E + Φ(u))
= elliptic integral if Φ(u) = u3
=⇒ ξ(u) =⇒ u(ξ)
yields results but no intuition
Dynamical systems approach with ξ as time
v ≡ dudξ
=⇒{
u̇ = v
v̇ = dΦdu− cv
If c = 0, then system is Hamiltonian:
H = −Φ + 12v2 =⇒
{u̇ = ∂H
∂v
v̇ = −∂H∂u
Add diffusion to supercritical pitchfork =⇒ Ginzburg-Landau equation:∂u
∂t= µu− u3 + ∂
2u
∂x2
Steady states
0 = µu− u3 + d2u
dx2
Integrate to obtain the potential:
−dΦdu
= µu− u3 =⇒ −Φ = µ2u2 − 1
4u4
Steady states:d2u
dx2=
dΦ
du=⇒
{u̇ = v
v̇ = dΦdu
Fixed points of new dynamical system:
0 = v
0 =dΦ
du= −µū + ū3 =⇒ ū = 0 or ū = ±√µ
Same ū as without diffusion, but stability under new dynamics is different:
J =(
0 1Φ′′ 0
)
=
(0 1
3ū2 − µ 0
)
=
(0 1−µ 0
)
or
(0 1
2µ 0
)
Hamiltonian⇐⇒ Tr(J ) = ∂2H∂u∂v− ∂2H
∂v∂u= 0⇐⇒ eigs are±λ
λ(−λ) = −Φ′′ =⇒ λ± = ±√Φ′′ =
{±√−µ for ū = 0±√2µ for ū = ±√µ
λ = ±iω =⇒ center = elliptic fixed pointλ = ±σ =⇒ saddle = hyperbolic fixed point
µ = −1 µ = +1
µ = +1
Types of Trajectoriesµ = −1 µ = +1
unbounded, crossing between left to right X X
unbounded, staying on left or on right X X
periodic X
front (limiting case of periodic) X
Periodic:
Trajectories in the (u, u̇) phase plane are elliptical.Particle oscillates back and forth in potential well.
Fronts:
Trajectory leaves ū = −√µ at zero velocity, arrives exactly at ū = √µwith zero velocity, since there is no friction.
Profile has u = −√µ on left, narrow transition region, u = √µ on right.
Type of trajectory is determined by the initial conditions (temporal point of
view) or the boundary conditions (spatial point of view). Periodic bound-
ary conditions on a domain of fixed wavelength select the periodic profile.
Boundary conditions u(±∞) = ±√µ lead to front solution.
Front solutions connect two maxima of −Φ, i.e. hyperbolic unstable fixedpoints of the transformed dynamical system.
These correspond to stable spatially homogeneous solutions to the original
reaction-diffusion system:du
dt= −dΦ
duStability determined by
−d2Φ
du2(ū)
{< 0> 0
}
=⇒ ū{
stable
unstable
}
Thus, homogeneous stable steady states are maxima of−Φ.
Nonzero c
Front between u−∞ and u+∞, which are maxima of −Φ(u) and hencestable solutions to spatially homogeneous equations,
Dirichlet BCs u(ξ = ±∞) = u±∞ =⇒ Neumann BCsdu
dξ(ξ = ±∞) = 0
Travelling wave solutions:
0 = cdu
dξ− dΦ
du+
d2u
dξ2
Multiply by du/dξ:
0 = c
(du
dξ
)2
− dΦ(u(ξ))dξ
+1
2
d
dξ
(du
dξ
)2
Integrate over ξ interval:
0 = c
∫ +∞
−∞dξ
(du
dξ
)2
−∫ +∞
−∞dξ
dΦ(u(ξ))
dξ+
∫ +∞
−∞dξ
1
2
d
dξ
(du
dξ
)2
= c
∫ +∞
−∞dξ
(du
dξ
)2
− [Φ]+∞−∞ +1
2
[(du
dξ
)2]+∞
−∞⇐ vanishes because of
Neumann BCs
c =Φ+∞ − Φ−∞∫ +∞−∞ dξ
(dudξ
)2 where Φ±∞ ≡ Φ(u±∞)
Front velocity c > 0 if Φ−∞ < Φ+∞, i.e. if−Φ−∞ > −Φ+∞.Front moves from left to right =⇒u−∞,−Φ−∞ domain invades u+∞,−Φ+∞ domainFront motion increases size of domain with greater−Φ.
Mechanical analogy:
Trajectory goes from u−∞, −Φ−∞ to u+∞, with lower potential −Φ+∞.For “velocity” du/dξ and “kinetic energy” to vanish at both endpoints,
energy must be lost via friction. Hence c is positive.
“Negative friction” is possible since c < 0 just means that the front movestowards the left.
Trajectory Phase portrait
from lower left hill to higher right hill Former center has become focus
uses “negative friction” to increase its energy surrounded by spiralling trajectories
Perturbed Ginzburg-Landau equation
0 = cdu
dξ+ µu− u3 − 0.1 + d
2u
dξ2
Potential
−Φ = 12µu2 − 1
4u4 − 0.1u
has two maxima of different heights