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| Both sides previous revision Previous revision Next revision | Previous revisionLast revisionBoth sides next revision | ||
| research_report_pix [2008-03-11 10:14] – 195.53.62.237 | research_report_pix [2008-03-12 10:56] – 195.53.62.237 | ||
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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, | ||
| + | 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 | ||
| Line 181: | Line 210: | ||
| 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, | My early plans were to make a neat, self contained application, | ||
| Line 199: | Line 225: | ||
| interface programming ([[http:// | interface programming ([[http:// | ||
| - | [ 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 | ||
| Line 221: | Line 246: | ||
| 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 | ||
| Line 274: | Line 298: | ||
| 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 | ||
| Line 348: | Line 371: | ||
| modifications to the Grid object provided by the | modifications to the Grid object provided by the | ||
| [[http:// | [[http:// | ||
| - | spread-sheet | + | spread-sheet |
| 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. |
| + | |||
| + | [ highf-grid.png ] | ||
| + | |||
| [ basin of attraction 0,0,0 assumption ] | [ basin of attraction 0,0,0 assumption ] | ||
| Line 372: | Line 399: | ||
| 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 ] | ||
nik