Mathematical Modelling Lecture 15 Fractals · Tools for constructing and manipulating models !...
Transcript of Mathematical Modelling Lecture 15 Fractals · Tools for constructing and manipulating models !...
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Introduction
Mathematical ModellingLecture 15 – Fractals
Phil [email protected]
Phil Hasnip Mathematical Modelling
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Introduction
Overview of Course
Model construction −→ dimensional analysisExperimental input −→ fittingFinding a ‘best’ answer −→ optimisationTools for constructing and manipulating models −→networks, differential equations, integrationTools for constructing and simulating models −→randomnessReal world difficulties −→ chaos and fractals
The material in these lectures may be found in Chaos UnderControl: The Art and Science of Complexity by D. Peak & MiM.Frame, pub. Freeman.
Phil Hasnip Mathematical Modelling
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Introduction
The story so far...
Last time we introduced the concept of fractal dimension, andshowed how we could be:
Measured experimentally using the box-counting methodCalculated analytically using the scaling relationship
S(bN) = bdS(N)
where S(N) is the length as measured by N rulers in eachdimension.
Phil Hasnip Mathematical Modelling
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Introduction
Recap: box-counting
Draw a grid with N intervals along each dimensioni.e. an N × N × . . .× N grid of boxesCount boxes needed to entirely contain shape, S(N)
Repeat for different N and either:Plot on a log-log graphPlot ln S(N) against ln N
Slope −→ fractional dimension df
Phil Hasnip Mathematical Modelling
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Introduction
Recap: analytical method
S(bN) = bdS(N)
where S(N) is the length as measured by N rulers in eachdimension.
Look at increase of quantity as increase sizeE.g. triple span of Koch curve −→ quadruple curve length⇒ 3d = 4⇒ d = ln 4
ln 3
Phil Hasnip Mathematical Modelling
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Introduction
The Sierpinski gasket
Phil Hasnip Mathematical Modelling
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Introduction
Key features
The non-integral dimension d is a measure of how a curveor shape fills up spaceSelf-similarity – as we change the magnification, the shapelooks similar−→ new detail emerges−→ no characteristic length scaleScale invariant
Phil Hasnip Mathematical Modelling
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Introduction
Key features
Fractal properties are described by power laws rather thanexponentials
Phil Hasnip Mathematical Modelling
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Introduction
Key features
Phil Hasnip Mathematical Modelling
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Introduction
Key features
Phil Hasnip Mathematical Modelling
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Introduction
Key features
Phil Hasnip Mathematical Modelling
![Page 12: Mathematical Modelling Lecture 15 Fractals · Tools for constructing and manipulating models ! networks, differential equations, integration Tools for constructing and simulating](https://reader033.fdocuments.us/reader033/viewer/2022050308/5f703a7fc55d9c401349a60e/html5/thumbnails/12.jpg)
Introduction
Key features
Phil Hasnip Mathematical Modelling
![Page 13: Mathematical Modelling Lecture 15 Fractals · Tools for constructing and manipulating models ! networks, differential equations, integration Tools for constructing and simulating](https://reader033.fdocuments.us/reader033/viewer/2022050308/5f703a7fc55d9c401349a60e/html5/thumbnails/13.jpg)
Introduction
Beyond simple Koch curves
These simple models still don’t look natural. What can we do?
Bit of randomness – several operations at each iteration,randomly select one self-affine
xn+1 = Txn + b
where T is a transformation, e.g. rotation, scaling,translation...Use full randomness – e.g. DLA.
Phil Hasnip Mathematical Modelling
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Introduction
Diffusion-limited aggregation
Phil Hasnip Mathematical Modelling
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Introduction
Mandelbrot
zn+1 = z2n + c
C(x , y) = x + iy (point on Argand diagram)z0 = 0Iterate for no, iterationsWhat value does zn tend to?If zn converges, plot as blackIf not converge, colour according to rate of divergenceLinks to chaos and population modelling – see next lecture!
Phil Hasnip Mathematical Modelling
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Introduction
Summary
Fractals have unusual scaling properties −→ fractionaldimensionsCommon in natureEasy to constructLinks to chaos and nonlinear models
Phil Hasnip Mathematical Modelling