Morphology of Experimental and Simulated Turing Patterns

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1.3 Morphological analysis with Minkowski functionalsA quantitative approach to characterize and compare experimental and numericalpatterns is given by the so-called Minkowski functionals. Minkowski functionals aremeasures from integral geometry and can by used to characterize the morphologyof greyscale images [38]. Minkowski functionals are useful to describe irregularstructures, as they give a measure of the connectivity of domains in the image.Recent applications include the characterization of spinodal decompositions in thinpolymer films during dewetting processes in Ref. [35] and the characterization ofstructure formation in thin films of cylinder forming block copolymers [51].A Minkowski analysis for the experimental patterns has been done by Mecke [37],who showed that the transition of experimental patterns in the CIMA reaction can bedescribed by symmetry breaking in certain coefficients that describe the functionalform of the Minkowski functionals. The analysis in this thesis is an application ofthis method to experimental and simulated Turing patterns.1.4 Pattern formation in other systemsApart from the CIMA reaction, only two other chemical systems which producestationary patterns under laboratory conditions have been found. One is the iodateferrocyanide-sulfite(FIS) reaction, which produces patterns under finite amplitudeperturbations, see Ref. [24], and the recently discovered thiourea-iodate-sulfite (TuIS)reaction in Ref. [19], where a general design scheme for pattern forming chemicalreactions is proposed. Another non-biological system with Turing pattern formationwas reported in silver/antimony electrodeposists on surfaces [56].Following Turing’s original idea reaction-diffusion equations are used to describepattern formation in biological organisms such as hydra in Ref. [15] or the spreadof brain tumors in Ref. [44] and bacterial patterns as described in Ref. [62] andshown in Fig. 1.4a. Furthermore reaction-diffusion equations can successfully modelspatial population dynamics in predator-prey systems, where finite-size effects giverise to additive noise that influences pattern formation, such as shown in Fig. 1.4band analyzed in recent work by Reichenbach et al. [52].Nonlinear pattern formation can also be seen in ferromagnetic fluids in externalmagnetic fields, known as Rosensweig-instability [8]. Other mechanisms of patternformation include self-organization of block co-polymers, which separate into distinctdomains that can form ordered patterns or the formation of fractal branch structuresby diffusion-limited aggregation [5].Another mechanism of pattern formation was found in vertically shaken granularmedia, as reported in Ref. [39]. A similar system with a variety of patterns, such asshown in Fig. 1.5, is analyzed in this thesis. It is the simplicity of the experimentalsystem, which only consists of a thin layer of beads and a driving plate, which makesthem particularly interesting.14
(a) “Bacteria” in a reactiondiffusionmodel forming concentricrings with region of high cell density(white) and low cell density (black).(Image reproduced from Ref. [62])(b) Steady spiral state in a noisythree species cyclic competitionmodel. Each color represents oneof the three species. (Image reproducedfrom Ref. [52])Figure 1.4: Pattern formation in ecological systems, such as bacteria and species competition.Such systems can be described by reaction-diffusion equations, which show spatialpatterns formation.Figure 1.5: Square and stripe patterns in vertically shaken granular media, see chapter 9.15
Page 5: AbstractTuring patterns formed in t
Page 10: 5.5 Minkowski-maps of patterns in t
Page 13: of the experimental quasi-2D patter
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 and 48: 1.61.41.21b0.80.60.40.205 10 15 20
Page 49: Figure 4.10: The concentration prof
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 =
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(a) Experimental hexagonalpattern(b
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5.4 Local Minkowski functionals of
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P V3210-1-2-1 -0.5 0 0.5 1ρP S2.42
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(a) Experimental hexagonalpattern(b
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5.5 Minkowski-maps of patterns in t
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V(ρ)10.80.60.40.200 0.2 0.4 0.6 0.
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(a) Experimental hexagonalpattern.
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5.7 Minkowski functionals of patter
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Chapter 6Influence of additive nois
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∆u/N 20.2∆u/N 20.1∆u/N 20.10.
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(a) Experimental lamellae.(b) Numer
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Chapter 7Statistical ensemble of su
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HexagonalLamellar0.10.20.51.0amplit
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Experimental patternSuperposed patt
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In this model no dynamics are expli
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Experimental patternSuperposed patt
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7.3 Interacting pattern gasIn this
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7.3.3 Numerical resultsIn this sect
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Chapter 8Summary and outlookA quant
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•••••••••••
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(a)(b)(c)(d)(e)(f)Figure 9.3: Patte
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Appendix ADetails on experimental d
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Bibliography[1] K. Agladze, E. Dulo
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[28] I. Lengyel and J. A. Pojman. A
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[54] B. Rudovics, E. Barillot, P. W
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ErklärungHiermit bestätige ich, d
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Morphology of Experimental and Simulated Turing Patterns

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