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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
2DPG 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
3DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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
4DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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
5DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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 )
6DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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,
7DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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,
8DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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
9DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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
10DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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]
11DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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
12DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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.
13DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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.
14DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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
15DPG Spring Meeting, SKM, Dresden DY 35.4 03.04.2014
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