Simulation of confined pocket of a city by leapfrogging method

Abstract.Leapfrogging appears at several levels and hence may be one of the fundamental processes generating urban fractality. Simulations have been performed on random pockets (slums) with different areas and lifetime of the Delhi city. The images were procured and processed using GIS package SAS planet and ImageJ and the fractal dimension was calculated by the box-counting method. In addition, we propose a confined leapfrogging model. Both studies suggests that any such confined pocket shall eventually have a high fractal dimension in the range 1.65-1.85 with a central tendency of 1.76 from the data of pockets and 1.78 from the confined model. These slums will approach a high fractal dimension due to the high level of social and economic pressure there. The confined leapfrogging model is suggested as the underlying process of slum growth.

[1]  Realizing the box-counting method for calculating fractal dimension of urban form based on remote sensing image , 2009 .

[2]  Tamás Vicsek,et al.  Fractal models for diffusion controlled aggregation , 1983 .

[3]  A. Jonkers.  Long-range dependence in the Cenozoic reversal record , 2003 .

[4]  M. Batty.  New ways of looking at cities , 1995, Nature.

[5]  Lianmao Peng,et al.  Controlling Molecular Growth between Fractals and Crystals on Surfaces. , 2015, ACS nano.

[6]  Lucien Benguigui,et al.  When and Where is a City Fractal? , 2000 .

[7]  L. Amaral,et al.  Multifractality in human heartbeat dynamics , 1998, Nature.

[8]  P. K. Leung,et al.  Self-similar fragmentation regulated by magnetic fields in a region forming massive stars , 2015, Nature.

[9]  H. Stanley,et al.  Multifractal phenomena in physics and chemistry , 1988, Nature.

[10]  H. Stanley,et al.  Magnitude and sign correlations in heartbeat fluctuations. , 2000, Physical review letters.

[11]  B. Mandelbrot.  How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , 1967, Science.

[12]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[13]  Steve McKillup,et al.  Statistics Explained: Contents , 2011 .

[14]  Bin Jiang,et al.  The fractal nature of maps and mapping , 2014, Int. J. Geogr. Inf. Sci..

[15]  Walter Willinger,et al.  Stock market prices and long-range dependence , 1999, Finance Stochastics.

[16]  Michalis Faloutsos,et al.  Long-range dependence ten years of Internet traffic modeling , 2004, IEEE Internet Computing.

[17]  A. Kolmogorov.  A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number , 1962, Journal of Fluid Mechanics.

[18]  Desheng Liu,et al.  Box-Counting Dimension of Fractal Urban Form: Stability Issues and Measurement Design , 2012, Int. J. Artif. Life Res..

[19]  Fractal features of urban morphology and simulation of urban boundary , 2008 .

[20]  L Benguigui,et al.  A Fractal Analysis of the Public Transportation System of Paris , 1995 .

[21]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[22]  Bin Jiang,et al.  The Evolution of Natural Cities from the Perspective of Location-Based Social Media , 2014, Digital Social Networks and Travel Behaviour in Urban Environments.

[23]  Peter H. Richter,et al.  The Beauty of Fractals , 1988, 1988.

[24]  Larry S. Liebovitch,et al.  Fractals and Chaos Simplified for the Life Sciences , 1998 .

[25]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[27]  David N. Kennedy,et al.  Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis , 2009, Brain Imaging and Behavior.

[28]  S. Havlin,et al.  Self-similarity of complex networks , 2005, Nature.

[29]  Hans Lauwerier,et al.  Fractals: Endlessly Repeated Geometrical Figures , 1991 .

[30]  R. Bailey,et al.  Quantitative evidence for the fractal nature of the stratigraphie record: results and implications , 2005 .

[31]  S. Pruess.  Some Remarks on the Numerical Estimation of Fractal Dimension , 1995 .

[32]  Modest von Korff,et al.  Molecular Complexity Calculated by Fractal Dimension , 2019, Scientific Reports.

[33]  Pierre Frankhauser,et al.  Aspects fractals des structures urbaines , 1990 .

[34]  Lucien Benguigui,et al.  City Growth as a Leap-frogging Process: An Application to the Tel-Aviv Metropolis , 2001 .

[35]  H. Stanley,et al.  Modelling urban growth patterns , 1995, Nature.

[36]  Tilmann Gneiting,et al.  Stochastic Models That Separate Fractal Dimension and the Hurst Effect , 2001, SIAM Rev..

[37]  Benoit B. Mandelbrot,et al.  Fractals and Chaos , 2004 .

[38]  W. Schlager.  Fractal nature of stratigraphic sequences , 2004 .

[39]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos with Student Solutions Manual , 2016 .

[40]  N. Rashevsky.  Life, information theory, and topology , 1955 .

[41]  Serge Salat,et al.  Assessing Urban Complexity , 2010 .

[42]  B. Mandelbrot,et al.  Fractal character of fracture surfaces of metals , 1984, Nature.

[43]  Zonglu He,et al.  Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications , 2018, Scientific Reports.

[44]  Walter Willinger,et al.  On the self-similar nature of Ethernet traffic , 1993, SIGCOMM '93.

[45]  Michael Batty,et al.  Fractal-Based Description of Urban Form , 1987 .

[46]  F. Hausdorff.  Dimension und äußeres Maß , 1918 .

[47]  Steve McKillup,et al.  Statistics Explained: An Introductory Guide for Life Scientists , 2006 .

[48]  Demetris Koutsoyiannis,et al.  Climate change, the Hurst phenomenon, and hydrological statistics , 2003 .

[49]  Michael Batty,et al.  Fractal Cities: A Geometry of Form and Function , 1996 .

[50]  S. Arlinghaus.  Fractals Take a Central Place , 1985 .