Arc-Fractal and the Dynamics of Coastal Morphology

Huynh, Hoai Nguyen; Chew, Lock Yue

doi:10.1142/s0218348x11005178

Cited by 7 publications

(9 citation statements)

References 29 publications

(25 reference statements)

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“…so that on the basis of Eq. (8) a = 0.654(6) and b = 0.692(12) or a = 0.60(4) and b = 0.80(8) depending on c (s) 1 . Fitting the data inTable VIagainst Eq.…”

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confidence: 99%

Abelian Manna model in three dimensions and below

Huynh¹,

Pruessner²

2012

Phys. Rev. E

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The Abelian Manna model of self-organized criticality is studied on various three-dimensional and fractal lattices. The exponents for avalanche size, duration, and area distribution of the model are obtained by using a high-accuracy moment analysis. Together with earlier results on lower-dimensional lattices, the present results reinforce the notion of universality below the upper critical dimension and allow us to determine the coefficients of an ε expansion. By rescaling the critical exponents by the lattice dimension and incorporating the random walker dimension, a remarkable relation is observed, satisfied by both regular and fractal lattices.

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“…so that on the basis of Eq. (8) a = 0.654(6) and b = 0.692(12) or a = 0.60(4) and b = 0.80(8) depending on c (s) 1 . Fitting the data inTable VIagainst Eq.…”

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confidence: 99%

Abelian Manna model in three dimensions and below

Huynh¹,

Pruessner²

2012

Phys. Rev. E

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“…Thirdly, scaling relations that are derived in a straightforward fashion on hypercubic lattices can be put to test in a more general setting. In this Brief Report, we address the first and the third aspect, by examining both numerically and analytically the scaling behaviour of the Abelian version of the Manna model [7][8][9] on two different fractal lattices.The fractal lattices used in this study are generated from the arc-fractal system [10]. The lattice sites are the invariant set of points of the arc-fractal.…”

mentioning

confidence: 99%

“…The fractal lattices used in this study are generated from the arc-fractal system [10]. The lattice sites are the invariant set of points of the arc-fractal.…”

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confidence: 99%

Abelian Manna model on two fractal lattices

2010

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We analyze the avalanche size distribution of the Abelian Manna model on two different fractal lattices with the same dimension dg = ln 3/ ln 2, with the aim to probe for scaling behavior and to study the systematic dependence of the critical exponents on the dimension and structure of the lattices. We show that the scaling law D(2 − τ ) = dw generalizes the corresponding scaling law on regular lattices, in particular hypercubes, where dw = 2. Furthermore, we observe that the lattice dimension dg, the fractal dimension of the random walk on the lattice dw, and the critical exponent D, form a plane in 3D parameter space, i.e. they obey the linear relationship D = 0.632(3)dg + 0.98(1)dw − 0.49(3). Although extensive research has been performed on self-organized criticality [1] for models on hypercubic lattices, far less work has been done on fractal lattices [2,3]. It remains somewhat unclear what to conclude from the latter studies. Fractal lattices are important for the understanding of critical phenomena for a number of reasons. Firstly, results for critical exponents in lattices with non-integer dimensions might provide a means to determine the terms of their = 4 − d expansion. Secondly, fractal lattices are particularly suitable for a real space renormalization group procedures, in particular that by . Thirdly, scaling relations that are derived in a straightforward fashion on hypercubic lattices can be put to test in a more general setting. In this Brief Report, we address the first and the third aspect, by examining both numerically and analytically the scaling behaviour of the Abelian version of the Manna model [7][8][9] on two different fractal lattices.The fractal lattices used in this study are generated from the arc-fractal system [10]. The lattice sites are the invariant set of points of the arc-fractal. We consider nearest neighbor interactions among sites. Here, the nearest neighbors of a given site are all sites which have the (same) shortest Euclidean distance to it. Our fractal lattices have no natural boundary; instead, they have only two end points at which two copies can join to form a bigger lattice. The dimension of the lattices is the same as the arc-fractal that generates them.In this study, we shall consider two fractal lattices: the Sierpinski arrowhead and the crab (see Fig. 1). The former is named "Sierpinski arrowhead" because it is the same as the well-known Sierpinski arrowhead [11], whereas the latter is termed "crab" because the overall shape of the generated lattice looks like a crab. These fractal lattices are generated through the arc-fractal system with number of segments n = 3 and opening angle of the arc α = π. For the Sierpinski arrowhead, the rule for orientating the arc at each iteration is "in-out-in", while the rule is "out-in-out" for the crab. Both lattices have the same dimension d g = ln 3/ ln 2 ≈ 1.58. The total number of sites on the lattice at the i-th iteration is N i = 3 i + 1. The coordination number of sites on these lattices varies between two and three. Asy...

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“…The arc-fractal system, which was introduced in [ 26 , 27 ], is a simple yet universal fractal generator by applying recursive operations on geometrical arcs. At every step of iteration, every arc is divided into a number of segments and each segment is replaced by a new arc (see Fig.…”

Section: Introductionmentioning

confidence: 99%

“…In this section, we shall describe the rules for labelling an arc element based on its orientation and also discuss on the basic features of the labels with respect to the arc-fractal system. The labelling rules were discussed in [ 26 ] and are summarised here for readability of the readers. These rules hold for the arc-fractal system with single-rule iteration, i.e .…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Sequences Generated by the Arc-Fractal System

2015

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We study properties of the symbolic sequences extracted from the fractals generated by the arc-fractal system introduced earlier by Huynh and Chew. The sequences consist of only a few symbols yet possess several nontrivial properties. First using an operator approach, we show that the sequences are not periodic, even though they are constructed from very simple rules. Second by employing the ϵ-machine approach developed by Crutchfield and Young, we measure the complexity and randomness of the sequences and show that they are indeed complex, i.e. neither periodic nor random, with the value of complexity measure being significant as compared to the known example of logistic map at the edge of chaos. The complexity and randomness of the sequences are then discussed in relation with the properties of associated fractal objects, such as their fractal dimension, symmetry and orientations of the arcs.

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Arc-Fractal and the Dynamics of Coastal Morphology

Cited by 7 publications

References 29 publications

Abelian Manna model in three dimensions and below

Abelian Manna model in three dimensions and below

Abelian Manna model on two fractal lattices

The Complexity of Sequences Generated by the Arc-Fractal System

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