Morphology of Experimental and Simulated Turing Patterns

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4.1.2 Oscillatory solutionsClose to the small region of the parameter space in Fig. 4.1, where the Hopf bifurcationoccurs for higher values of b than the Turing bifurcation, i.e. from a 1 ≈ 6.455 toa 2 ≈ 8.984, the system shows spatial homogeneous oscillatory solutions. The evolutionsfrom a noisy initial state to a homogeneous oscillatory state can be visualizedby plotting the average concentration 〈u〉 of a solution over the simulation time asshown in Fig. 4.9. The time evolution of ∆u shows how the system converges to ahomogeneous state.Oscillatory hexagonal patterns, such as obtained in Ref. [53] in a normal-form analysis,were not found for direct numerical integration of the LE model. As reported inRef. [42] oscillatory hexagonal spots can only be found in the CIMA reaction with aconstant background illumination (the CIMA reaction is a photosensitive reaction)as a further control parameter and numerically a much smaller value of σ has to beassumed.2.221.81.6∆u10.10.010.0010.00011e-051e-061e-071.41.20 100 200 300 400 500t(a) Oscillating 〈u〉.1e-081e-091e-100 50 100 150 200 250 300 350 400 450 500t(b) Decreasing ∆u.Figure 4.9: The system converges to a homogeneous oscillatory solution in the parameterspace of the LE model at (a, b) = (8.784, 0.118).4.2 Concentration fluctuations of stable homogeneousstatesWhen plotting the patterns with a local greyscale, i.e. with maximum and minimumconcentrations of each pattern corresponds to black and white, also the numericalsolutions above the Turing bifurcation show patterns. This results in patternsolutions with very small amplitude for parameters above the Turing bifurcation.Note that the same greyscale values in each picture correspond to very differentconcentrations.For values of b above the Turing bifurcation the range of concentrations is so small,as indicated in Fig. 4.5b that the patterns seen in the numerical solutions can be con-48
Figure 4.10: The concentration profiles in the analyzed parameter region with a localgreyscale. Turing (red) and Hopf bifurcation (green) from Fig. 4.1 are also shown.sidered merely as artefacts of the non-homogeneous initial conditions. The changeof concentrations is at least 10 −8 × 〈u〉 smaller, with the average concentration 〈u〉.Given the numerical accuracy of our finite-difference method, the very low intensitypatterns above the Turing bifurcation cannot be rigorously analyzed. It is unclear ifthey are all numerical artefacts or the result of the nonlinearity of the reaction termsin the LE model. In accordance with Ref. [54] they are considered as homogeneoussolutions.49
Page 5: AbstractTuring patterns formed in t
Page 10: 5.5 Minkowski-maps of patterns in t
Page 13 and 14: of the experimental quasi-2D patter
Page 15: (a) “Bacteria” in a reactiondif
Page 18 and 19: Figure 2.1: First evidence for Turi
Page 20 and 21: 2.2 Lengyel-Epstein (LE) model for
Page 22 and 23: After rescaling the differential eq
Page 24 and 25: A full derivation is given in Ref.
Page 27 and 28: Chapter 3Numerical solution of 2Dre
Page 29 and 30: The advantage of the FTCS method is
Page 31 and 32: andA u · u n+1 = b n u (u,v) (3.14
Page 33 and 34: withχ u = 2γ u sin 2 k x∆2+ 2γ
Page 35 and 36: 3.4 Convergence to stationary patte
Page 37 and 38: 3.5 Discretization of stochastic pa
Page 39 and 40: Chapter 4Patterns in the Lengyel-Ep
Page 41 and 42: (a) Inverted hexagonalstate: a = 9.
Page 43 and 44: Figure 4.4: Overview of pattern for
Page 45 and 46: 1.61.41.21b0.80.60.40.205 10 15 20
Page 47: 1.61.41.21b0.80.60.40.205 10 15 20
Page 52 and 53: (a) ρ = 0.22, V = 0.77, S =0.15,
Page 54 and 55: Figure 5.2: Illustration for the ma
Page 56 and 57: perimeter S(ρ) and the Euler chara
Page 58 and 59: (a) Experimental hexagonalpattern.
Page 60 and 61: 5.3 Minkowski functionals of simula
Page 62 and 63: (a) Numerical hexagonalpattern, b =
Page 64 and 65: (a) Experimental hexagonalpattern(b
Page 66 and 67: 5.4 Local Minkowski functionals of
Page 68 and 69: P V3210-1-2-1 -0.5 0 0.5 1ρP S2.42
Page 70 and 71: (a) Experimental hexagonalpattern(b
Page 72 and 73: 5.5 Minkowski-maps of patterns in t
Page 74 and 75: V(ρ)10.80.60.40.200 0.2 0.4 0.6 0.
Page 76 and 77: (a) Experimental hexagonalpattern.
Page 78 and 79: 5.7 Minkowski functionals of patter
Page 81 and 82: Chapter 6Influence of additive nois
Page 83 and 84: ∆u/N 20.2∆u/N 20.1∆u/N 20.10.
Page 85 and 86: (a) Experimental lamellae.(b) Numer
Page 87 and 88: Chapter 7Statistical ensemble of su
Page 89 and 90: HexagonalLamellar0.10.20.51.0amplit
Page 91 and 92: Experimental patternSuperposed patt
Page 93 and 94: In this model no dynamics are expli
Page 95 and 96: Experimental patternSuperposed patt
Page 97 and 98: 7.3 Interacting pattern gasIn this
Page 99 and 100:
7.3.3 Numerical resultsIn this sect
Page 101 and 102:
60000040000050000040000035000030000
Page 103 and 104:
Chapter 8Summary and outlookA quant
Page 105 and 106:
•••••••••••
Page 107 and 108:
(a)(b)(c)(d)(e)(f)Figure 9.3: Patte
Page 109 and 110:
Appendix ADetails on experimental d
Page 111 and 112:
Bibliography[1] K. Agladze, E. Dulo
Page 113 and 114:
[28] I. Lengyel and J. A. Pojman. A
Page 115 and 116:
[54] B. Rudovics, E. Barillot, P. W
Page 117:
ErklärungHiermit bestätige ich, d
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Morphology of Experimental and Simulated Turing Patterns

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