Morphology of Experimental and Simulated Turing Patterns

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AbstractTuring patterns formed in the chlorite-iodide-malonic acid (CIMA) reaction, a farfrom equilibrium chemical reaction-diffusion system, are morphologically comparedto patterns obtained in the Lengyel-Epstein (LE) model, which is based on a nonlinearpartial differential equation and believed to model the CIMA reaction.A quantitative morphological analysis via Minkowski functionals, inspired by thework of Mecke (PRE, 53(5)), shows significant differences between the concentrationprofiles of patterns in the CIMA reaction and the LE model. This indicatesthat the deterministic LE model, while reproducing the basic character of the CIMAreaction, does not reproduce the correct concentration profiles.The concentration profiles measured in the experiment are given as 2D greyscaleimages for the CIMA reaction. The LE concentration profiles are obtained by solvingthe corresponding reaction-diffusion equation numerically using a finite-differencemethod.Minkowski functionals are morphological measures from integral geometry and characterizethe morphology of greyscale images. Our analysis shows that the CIMA andLE patterns differ in all Minkowski functionals, when extended fractions of patternsare analyzed. However a local agreement between crystalline, i.e. more ordered partsof the experimental patterns, and the patterns obtained in the LE model is found.We have excluded the possibility that additional additive white noise in the reactiondiffusionequations leads to an agreement of numerical and experimental patterns.An extended model based on the statistical superposition of basic patterns obtainedfrom the LE model produces patterns with a good morphological agreement tothe experimental hexagonal and lamellar Turing patterns. It also reproduces, forthe first time, the morphology of the turbulent phase, identified by Ouyang andSwinney (Chaos, 1(4)). This indicates that turbulent patterns could be describedas a dynamically fluctuating superposition of basic patterns.5
Page 9 and 10: Contents1 Introduction 111.1 Turing
Page 12 and 13: (a) Numerical hexagonalpattern(b) N
Page 14 and 15: 1.3 Morphological analysis with Min
Page 17 and 18: Chapter 2Turing patterns in the CIM
Page 19 and 20: gel diskABgellightClO − 2I − ,
Page 21 and 22: in the gel. This mechanism results
Page 23 and 24: steady state will either diverge or
Page 25: Linear stability analysis distingui
Page 28 and 29: t∆y∆xFigure 3.1: Distretized sp
Page 30 and 31: j,l-1,n+1j-1,l,n+1j,l,n+1j+1,l,n+1j
Page 32 and 33: 3.3 Von Neumann stability analysisT
Page 34 and 35: The ansatz from Eq. (3.24) yields[
Page 36 and 37: To illustrate the time evolution of
Page 38 and 39: where N(0,1) denotes the normal dis
Page 40 and 41: 1.61.41.2b Turing1b0.80.60.4b Hopf0
Page 42 and 43: (a) Hexagonal pattern for u(b) Hexa
Page 44 and 45: Average concentration b1.61.41.210.
Page 46 and 47: 1.61.41.21b0.80.60.40.205 10 15 20
Page 48 and 49: 4.1.2 Oscillatory solutionsClose to
Page 51 and 52: Chapter 5Morphology of experimental
Page 53 and 54:
threshold makes Minkowski functiona
Page 55 and 56:
Greyscale of experimental patternsA
Page 57 and 58:
(a) Experimental hexagonalpattern.
Page 59 and 60:
Table 5.1: Coefficients for the pol
Page 61 and 62:
olds, where the image is filled by
Page 63 and 64:
(a) Numerical hexagonalpattern, b =
Page 65 and 66:
(a) Experimental lamellar pattern(b
Page 67 and 68:
V(ρ)10.80.60.40.200 0.2 0.4 0.6 0.
Page 69 and 70:
V(ρ)10.80.60.40.200 0.2 0.4 0.6 0.
Page 71 and 72:
(a) Experimental hexagonalpattern(b
Page 73 and 74:
10.20.006V(ρ)0.80.60.4S(ρ)0.150.1
Page 75 and 76:
5.6 Minkowski analysis of the CDIMA
Page 77 and 78:
(a) Experimental hexagonalpattern.
Page 79:
V(ρ)10.80.60.40.200 0.2 0.4 0.6 0.
Page 82 and 83:
inverted hex.a = 9.4b = 0.06lamella
Page 84 and 85:
(a) Experimental hexagon.(b) Numeri
Page 86 and 87:
(a) Experimental lamellae.(b) Numer
Page 88 and 89:
(a) A square of size N/ √ 2×N/
Page 90 and 91:
7.2 Minkowski analysis for superpos
Page 92 and 93:
Table 7.2: The fit coefficients for
Page 94 and 95:
Experimental patternSuperposed patt
Page 96 and 97:
Experimental patternSuperposed patt
Page 98 and 99:
7.3.2 Second order approximationFor
Page 100 and 101:
(a) Patterns in the parameter space
Page 102 and 103:
102
Page 104 and 105:
104
Page 106 and 107:
Figure 9.2: A cardboard stripe atta
Page 108 and 109:
(a) (b) (c)Figure 9.4: Flat surface
Page 110 and 111:
110
Page 112 and 113:
[13] P. Evesque and J. Rajchenbach.
Page 114 and 115:
[41] W. Mickel, S. Münster, L. M.
Page 116 and 117:
[67] L. Yang, M. Dolnik, A. M. Zhab
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Morphology of Experimental and Simulated Turing Patterns

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