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research_report_pix [2008-03-11 10:14] 195.53.62.237research_report_pix [2013-08-22 10:28] (current) – [References] nik
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 problems of how to sensibly navigate or visualise such high dimensional spaces. problems of how to sensibly navigate or visualise such high dimensional spaces.
  
 +As an aside, many of the plots in this report are of my first automatically
 +discovered attractor which uses degree 5 polynomials. This means that all terms
 +are represented in which the exponents of the X, Y and Z variables sum to 5 or
 +less. This results in 3 equations of 56 terms, making for 168 parameters (or
 +170 if you count the 3 starting values of X, Y and Z).
  
 +Normally, attractors are found within this vast numerical space with a
 +combination of brute force (trying many different random sets of parameters)
 +and automated analysis to determine when an interesting attractor has been
 +found. The two analysis methods employed in the program presented in CPiC are
 +measurements of the correlation dimension and the Lyapunov exponent.
 +
 +The correlation dimension is a particular was of measuring fractal dimension,
 +which is a method of measuring the way in which fractal objects fill space. In
 +the case of the dot plots of strange attractors, the correlation dimension can
 +indicate if the attractor is a collection of disconnected points (dimsions
 +close to 0), if the points are arranged in the form of a line (dimension close
 +to 1), if the points are spread out into a flat plane (dimension close to 2) or
 +if the points form a voluminous cloud (dimension close to 3). Interesting
 +attractors tend to have a dimension greater than 1. Correctly measuring the
 +correlation dimension requires too many calculations to be feasible. As an
 +alternative, the correlation dimension is normally measured approximately using
 +a random sample of points. The accuracy of the measurement increases with the
 +number of points being tested. Additionally the dimension of a fractal is often
 +not consistent across the whole of the fractal, and the resulting value is only
 +an average of the dimension across the tested points.
 +
 +The Lyapunov exponent is a measure of the chaotic behaviour of the fractal.
 +Chaos is concerned with sensitivity of a complex system to small changes in
 +initial conditions. The Lyapunov measures the speed at which slightly different
 +starting conditions diverge.
  
-[ automated help looking at the space, fractal dimension, lyapunov ] 
  
 It is hoped that the ability to explore and derive a structural understanding It is hoped that the ability to explore and derive a structural understanding
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 face was the construction of a suitable interface for conveniently navigating face was the construction of a suitable interface for conveniently navigating
 the high dimensional number spaces.  the high dimensional number spaces. 
- 
-[ display issues ... when did i switch to soya? .. not documented, just before 
-getting the dot-plot working i guess ] 
  
 My early plans were to make a neat, self contained application, displaying the My early plans were to make a neat, self contained application, displaying the
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 interface programming ([[http://www.python.org/|Python]]). interface programming ([[http://www.python.org/|Python]]).
  
-[ summary of the problem? ] 
  
 After a number of different trial and error approaches to the problem, I gave After a number of different trial and error approaches to the problem, I gave
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 generating maps of the parameter space would be much easier than I had generating maps of the parameter space would be much easier than I had
 imagined. In fact, I was able to render my first parameter maps long before I imagined. In fact, I was able to render my first parameter maps long before I
-had a working renderer for the 3D dot plots of the attractors themselves.+had a working renderer for the 3D dot plots of the attractors themselves.  
  
-[ randomly chosen attractor 168coeffs.. is that degree 3? ]  
  
 The first plots were quite time consuming. For each point on the map I was The first plots were quite time consuming. For each point on the map I was
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 being used resulted in a further 25% speed increase. being used resulted in a further 25% speed increase.
  
-[ # should this data heavy optimisation stuff appear in results? ] 
  
 Performance gains from each progressive optimisation were decreasing, and I was Performance gains from each progressive optimisation were decreasing, and I was
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 modifications to the Grid object provided by the modifications to the Grid object provided by the
 [[http://www.wxpython.org/|wxWidgets]] library, which provides a rudimentary [[http://www.wxpython.org/|wxWidgets]] library, which provides a rudimentary
-spread-sheet display of a grid of numbers. I added the ability to easily+spread-sheet interface. I added the ability to
 incrementally modify the number in the current cell by scrolling with the mouse incrementally modify the number in the current cell by scrolling with the mouse
-wheel.+wheel. Different combinations of the Alt, Shift and Control keys change the amount by which the value is incremented. 
 + 
 +[ highf-grid.png ] 
 + 
  
 [ basin of attraction 0,0,0 assumption ]  [ basin of attraction 0,0,0 assumption ] 
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 highf-100.png (accidental basin plot) highf-100.png (accidental basin plot)
  
-[ first dot plot ]+[ first dot plot: highf-dotplot.png ]
  
 [ low order ] [ low order ]
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 ==== References ==== ==== References ====
- 
   * http://www.rawilson.net/chaos/lyapunov/index.html   * http://www.rawilson.net/chaos/lyapunov/index.html
- 
   * http://sprott.physics.wisc.edu/sa.htm   * http://sprott.physics.wisc.edu/sa.htm
   * http://ibiblio.org/e-notes/MSet/Logistic.htm   * http://ibiblio.org/e-notes/MSet/Logistic.htm
-  +  * images and animations at [[Pix Strange Attractor]]
-  * images and animations at  [[Pix Strange Attractor]]+
  
  
  
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