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.
Phil Hasnip Mathematical Modelling
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
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
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
Introduction
The Sierpinski gasket
Phil Hasnip Mathematical Modelling
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
Introduction
Key features
Fractal properties are described by power laws rather thanexponentials
Phil Hasnip Mathematical Modelling
Introduction
Key features
Phil Hasnip Mathematical Modelling
Introduction
Key features
Phil Hasnip Mathematical Modelling
Introduction
Key features
Phil Hasnip Mathematical Modelling
Introduction
Key features
Phil Hasnip Mathematical Modelling
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
Introduction
Diffusion-limited aggregation
Phil Hasnip Mathematical Modelling
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
Introduction
Summary
Fractals have unusual scaling properties −→ fractionaldimensionsCommon in natureEasy to constructLinks to chaos and nonlinear models
Phil Hasnip Mathematical Modelling