Iteration of functions and fractal patterns

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Iteration of functions and fractal patterns

• Edith Perrier, IRD France,
• Visiting scientist at UCT,
• Applied Maths Dept., room 417.1, Ext. 3205,
• E-mail edith_at_maths.uct.ac.za
• Web site where to download previous lectures
  lect.ppt (or lect.zip if links included)
  http//www.mth.uct.ac.za/Affiliations/BioMaths/Ind
  ex.html

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A well known fractal pattern ! The middle third
Cantor Set
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Iterated functions

• A mapping fIR-gtIR, IR2-gtIR2,C-gtC, E-gtE
• Iterates of order k
• Example f IR-gtIR
• Let us look for the iterates of f and for the
  largest bounded invariant set under f
• f(Cantor Set)Cantor Set

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The logistic map

• Fatou (1906) for parameters values less than -2
  or more than 4, Fatou dusts similar to Cantor
  dusts
• Population dynamics
• Discrete form
• Iteration of f(x)2x(1-x) starting with x0.1
• 0.1 0.18 0.2952 0.41611392 0.4859262512
  0.4996038592 0.4999996862 0.5000000000
• For large values one can
  show (Falconer p.174) that the invariant set is
  fractal with a Hausdorff dimension

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Bifurcations diagram of the logistic map
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The squaring generators

• f IR-gtR
• Equivalence with the logistic map with
  appropriate coordinates change and
• f C-gtC
• For a complex number c, the filled-in Julia set
  of c is the set of all z for which the iteration
  of f(z )does not diverge to infinity. The Julia
  set is the boundary of the filled-in Julia set.
  For almost all c, these sets are fractals.
• Julia sets are usually fractal sets
• When IcI is large
• When IcI is small

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Iterations of z-gtz2

• 0
• 1
• i
• -2
• -2i

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Exercice

• Calculate J0
• Imagine Jc for very small c

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Solution

• Unit circle

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Mandelbrot set and Julia sets
F C-gtC Mandelbrot set
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IFS (Iterated Function Systems)

• Definition An iterated function system is a
  finite set
• of affine transformations of IR2
• Each affine transformation w (in two dimensions)
  is a function that performs some combination of
  scaling, rotation, and translation on points in a
  plane and takes the general form
• w (x,y) (ax by e , cx dy f )
• Its determined by six real numbers
• The notion of IFS applies to higher-dimensional
  spaces . See p.80. .M. F. Barnsley. Fractals
  everywhere. Academic Press, Boston, 2nd edition,
  1993.

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Examples

• Von Koch
• Sierpinski carpet
• From Larry Riddle, Department of Mathematics,
  Agnes Scott College http//ecademy.agnesscott.edu/
  lriddle/ifs/

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• An important class of processes that produce
  fractal patterns are random iteration algorithms,
  which produce images of fractal objects. The
  procedure is akin to using a pen to mark dots at
  random on a sheet of paper. However, instead of
  being completely random, the movement of the pen
  from one position to the next is selected, at
  random, from a set of rules, each having fixed
  probability of being chosen. We can define these
  rules by affine transformations.

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Examples

• ------------ Parameter a b c d e f p ----
• Fern
• 0.0 0.0 0.0 0.16 0.0 0.0 0.10
  
• 0.2 -0.26 0.23 0.22 0.0 1.6 0.08
• -0.15 0.28 0.26 0.24 0.0 0.44 0.08
• 0.75 0.04 -0.04 0.85 0.0 1.6 0.74
• Grass
• 0.0 0.0 0.0 0.5 0.0 0.0 0.15
• 0.02 -0.28 0.15 0.2 0.0 1.5 0.10
• 0.02 0.28 0.15 0.2 0.0 1.5 0.10
• 0.75 0.0 0.0 0.5 0.0 4.6 0.65
• --------------------------------------------------
  ----
• The 7th paramater is the probability p which
  assigns the frequency with which the
  transformation is used.
• http//life.csu.edu.au/complex/tutorials/tutorial3
  .html

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Lecture series

• Lecture 1. March. 3rd. Introduction to fractal
  geometry . Measures and power laws.
• Lecture 2. March 5th. Definitions of non-integer
  dimensions and mathematical formalisms
• Lecture 3. March 10th. Iteration of functions and
  fractal patterns
• Lecture 4. March 12th. Extensions Self-similar
  and self-affine sets . Multifractals.
• Lecture 5. March 17th. Fractals / Geostatistics
  / Time series analysis
• Lecture 6. March 19th. Dynamical processes
  Fractal and random walks
• Lecture 7. March 24th. Dynamical processes
  Fractal and Percolation
• Lecture 8. March 26th. Dynamical processes
  Fractal and Chaos

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