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Cours : Dynamique Non-Lin´ eaire Laurette TUCKERMAN [email protected] VII. Reaction-Diffusion Equations: 1. Excitability 2. Turing patterns 3. Lyapunov functionals 4. Spatial analysis and fronts

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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=

    du−c du

    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

    [

    −Φ + 12

    (du

    )2]

    = −dΦdu

    du

    dξ+

    du

    d2u

    dξ2

    =

    [

    −dΦdu

    +d2u

    dξ2

    ]du

    dξ= −c

    (du

    )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

    )2

    E + Φ(u(ξ)) =1

    2

    (du

    )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=

    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

    )2

    − dΦ(u(ξ))dξ

    +1

    2

    d

    (du

    )2

    Integrate over ξ interval:

    0 = c

    ∫ +∞

    −∞dξ

    (du

    )2

    −∫ +∞

    −∞dξ

    dΦ(u(ξ))

    dξ+

    ∫ +∞

    −∞dξ

    1

    2

    d

    (du

    )2

    = c

    ∫ +∞

    −∞dξ

    (du

    )2

    − [Φ]+∞−∞ +1

    2

    [(du

    )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