mathematical modelling lecture 15 fractals · tools for constructing and manipulating models !...

Introduction

Mathematical ModellingLecture 15 – Fractals

Phil [email protected]

Phil Hasnip Mathematical Modelling

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.


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.


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


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


Introduction

The Sierpinski gasket


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


Introduction

Key features

Fractal properties are described by power laws rather thanexponentials


Introduction

Key features


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.


Introduction

Diffusion-limited aggregation


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!


Introduction

Summary

Fractals have unusual scaling properties −→ fractionaldimensionsCommon in natureEasy to constructLinks to chaos and nonlinear models


mathematical modelling lecture 15 fractals · tools for constructing and manipulating models !...

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