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1DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014

Turing instability in a scalar reaction-diffusion system with delay

Andreas Otto, Jian Wang, Günter Radons

Complex Systems and Nonlinear DynamicsInstitute of Physics Chemnitz University of Technology

DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014


Introduction – scalar delayed reaction-diffusion systems

● Delay in reaction term f:

● e.g.: model of population growth with diffusion:

● Distributed delay with delay kernel (different delays for different individuals)

● Initial condition:

→ infinite dimensional in space and time

∂∂ tu( x , t )=f (u ( x , t) , uτ ( x , t ))+DΔu ( x , t ) , D>0

u τ( x , t)=∫min ( τ)

max (τ )

ρ(θ)u( x , t−θ)d θ

delay τ

Time to sexual maturity

ρ(θ)

u( x ,θ)=Φ( x ,θ) , −max( τ )⩽θ⩽0


Introduction – Turing instability [1]

[1] A.M. Turing: “The chemical basis of morphogenesis”, Phil. Trans. Roy. Soc. B, 237:37-72 (1952).

● Stable system without diffusion ● With diffusion → term with Laplace operator

→ Negative spectrum

→ Diffusion itself is ‘stable’

● But both together can be unstable:

Turing instability = diffusion-driven instability

σ(Δ )⊂ℝ0-

stablesystem diffusion

unstablesystem


Introduction – Turing instability [1]

[1] A.M. Turing: “The chemical basis of morphogenesis”, Phil. Trans. Roy. Soc. B, 237:37-72 (1952).

u=f (u , v )+ DuΔ uv=g(u , v )+ DvΔ v

unstable for some D

u, D

v > 0

→ stable system

● Only possible in multi-component systems:- sum of two matrices with only negative eigenvalues can be a matrix with a positive eigenvalue- sum of two negative scalars can give no positive scalar

● Example [1]:

● Homogeneous state with wavenumber k = 0 is stable (stable dynamics without diffusion)

● Any spatially inhomogeneous state with wavenumber k > 0 is unstable

→ Mechanism for pattern formation


Stability analysis – equilibrium

● Homogeneous equilibrium:● Infinitesimal perturbation around equilibrium u*:

● Coefficients depend on derivatives of reaction term f:

● Fourier transformation in space:

● Laplace transformation in time:

f (u* , u τ* )=0

∂∂ t

ξ( x , t )=(α+DΔ)ξ( x , t )+βξ τ( x , t)

ξ

α=∂ f (u ,uτ)

∂u ∣u=u* , uτ=u

*, β=

∂ f (u ,uτ)∂ uτ ∣

u=u* , uτ=u*

u( x , t )=u* ,

∂∂ t

ξ(k ,t )=(α−Dk 2)ξ(k , t)+βξ τ(k , t)

x → k

(α−D k 2+βρ(sk )−sk )ξ(k , sk )=0

t → sk , ρ(θ) → ρ( sk )


Stability analysis – equilibrium

Characteristic equation:

1. no delay ( ):● No Turing instability in a scalar system

- stable for the homogeneous mode with k = 0:

→ stability of inhomogeneous modes with k > 0:

2. discrete delay ( ):● Transcendental characteristic equation

● Infinitely many characteristic roots sk because of the delay

● Scalar system but infinite dimensional due to delay● Turing instability cannot be excluded

→ Further investigation of characteristic roots

sk=λk+ iωkα−D k 2+βρ( sk )−sk=0,

β=0 α−D k 2−sk=0

s0=λ0=α<0

sk=λk=α−Dk 2<0

ρ(θ)=δ(θ−τ) α−D k 2+βe−sk τ−sk=0,


Stability analysis – equilibrium, discrete delay

● Without loss of generality

● D-curves ( ) in the α-β-plane for k = 0 (no diffusion)● Infinitely many D-curves

due to transcendental characteristic equation

● Separation between stable and unstable parameters possible

● Turing instability:- stable for k = 0 (white area)- unstable for k > 0 (decreasing α)

→ no Turing instability for discrete delay

τ=1

λ0=0

α−Dk 2+βe−sk−sk=0,


Stability analysis – equilibrium, distributed delay

3. distributed delay: ● Example: asymmetrically distributed

(two delays, non-uniform weights)

A=0.52

stable

● D-curves ( ) in the α-β-plane for k = 0 (no diffusion)

● Non-monotonic behavior of D-curve, that separates stable from unstable behavior ( )

● Turing instability is possible:- stable for k = 0 and (white area)- unstable for k > 0 (decreasing α)

λ0=0


Pattern formation – distributed delay

● 2D Turing pattern for Fisher-KPP equation with distributed delay(delay distribution as in previous slide):

● Traveling wave solution (k and ω close to linear stability analysis)

x1

x2

u(x1,x

2,t)

f (u ,uτ )=a u(1−uτ)

α=0β=−6.1D=0.1


Conclusion

• Turing instability can occur in one-component reaction-diffusion systems with distributed delays

→ diffusion induced wave bifurcation

• Additional remarks:- Turing instability was not found for symmetrically distributed delays- Extension to time-varying delays possible: fast-varying delay → distributed delay → Turing instability possible

slowly-varying delay → sequence of discrete delays → no Turing instability

Thank you for your attention

Questions?Contact: [email protected]

mailto:[email protected]


Stability analysis – symmetrically distributed

● Constant distributed delay – symmetrically distributed around ● Example: - 2 delays, equally weighted

→ D-curves in the α-β-plane for k = 0:

τ0=1

● no Turing instability for any symmetric delay distribution was detected

● No proof for the absence of Turing instability for symmetrically distributed delay

stable


Stability analysis - slowly time-varying

2. Case: equilibrium + slowly time-varying delay

● Frozen time approach [6]:

→ Series of subsequent autonomoussystems with frozen delay

→ Characteristic equation for each of the P frozen delays τi = τ(ti)

● Stability is determined by

u τ( x , t )=u( x , t−τ(t ))

Time t

Time t

Per

turb

atio

n ξ(

0,t)

Del

ay τ

(t)

ski=α−Dk 2+ βe−sk

iτi

λk=1P∑i=1

P

λ ki

[6] A. Otto and G. Radons: “Application of spindle speed variation for chatter suppression in turning”, CIRP J. Manuf. Sci. Tech. 6:102-109 (2013).

u(t )=u* → α ,β=const.


Stability analysis - fast time-varying delay

3. Case: equilibrium + fast time-varying delay

● Equivalent to distributed delay with kernel of a moving delta peak:

● Fast time-varying delay:

→ Averaging of the delay distribution [7]

→ Autonomous system similar to case 1

u τ( x , t)=u( x , t−τ(t ))

u τ( x , t)=∫min ( τ)

max (τ )

δ(θ−τ(t))u( x , t−θ)d θ

ρ(θ)≈ limT →∞

1T ∫0

Tδ(θ−τ(t ))dt

[7] W. Michiels, V. V. Assche, S.I. Niculescu: “Stabilization of time-delay systems with a controlled time-varying delay and applications”, IEEE Trans. on Automatic Control 50:493-504 (2005).

ρ(θ)

θ

τ(t )

t

u(t)=u* → α ,β=const.


Results - slowly time-varying

● Slowly time-varying delay (large period T) ● Same parameters

(asymmetric delay variation)

α=0, β=−6.1

→ good agreement with theory(case 2, frozen time approach)

→ generally no Turing instability possible


Results - fast time-varying

● Fast time-varying delay (small period T) ● Example of the previous slide

→ good agreement with theory (case 3)

α=0, β=−6.1

stable


Results - equilibrium

● Arbitrarily time-varying delay (case 4) ● Transition between two limiting cases Slowly time-varying

(frozen time approach)

Fast time-varying(distributed delay)

α=0, β=−6.1

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