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Algorithms for Ploting Basins of Attraction of Rational Maps and Calculating Their Areas
Published on 2013-11-041754 Views
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Presentation
Summer School on Computational Topology and Topological Data Analysis00:00
Iterative root- finding methods and rational functions00:19
Pre-existing similar computational programs - 100:57
Pre-existing similar computational programs - 201:14
Objectives01:24
Visualizing basins of attraction01:47
Approximating areas of basins of attraction02:42
Discrete semi-flows03:20
Fixed and periodic points04:04
Smooth and complex structures04:14
Bijections among P104:42
Chordal metric on P104:54
Normalized homogeneous coordinates05:20
Complex rational maps05:44
End points associated to a discrete semi-flow06:33
Basins of end points07:40
Fixed points of a rational map08:42
Rational functions induced by iterative numerical methods10:01
Discrete semi-flows induced by a rational map10:52
Graphic representation of basins of attraction11:35
Calculation of the fixed points12:21
Distance between two points13:03
Iteration of the rational map13:31
Determination of the fixed point14:40
Basins of n- cycles of a rational map15:15
Color assignment strategies16:12
Fixed point to which the iteration sequence converges17:07
Number of iterations until convergence17:15
Combination of the both previous strategies17:21
Plotting algorithms17:48
FractalPlot19:02
FractalPlotInsideOutside19:05
SpherePlot19:12
SpherePlot vs cubicSpherePlot19:17
Projection of subdivisions: subdivision of a spherical surface19:35
Projection of subdivisions: obtaining iteration points and classifying regions - 120:17
Projection of subdivisions: obtaining iteration points and classifying regions - 220:52
Subdivisions of the projection - 121:09
Subdivisions of the projection - 221:37
Area of a spherical quadrilateral22:04
Angles of a spherical polygon22:53
Calculation of tangents23:21
Probability associated to a root and example of use23:28
Connections with Fractal Geometry24:39
Application of the algorithm for calculating areas of basins25:43
Comparison between areas: lowest multiplicity fixed26:23
Potential curve fi tting - 126:52
Potential curve fi tting - 227:04
Comparison between areas: highest multiplicity fixed27:06
Polynomial curve fitting - 127:31
Polynomial curve fitting - 227:34
Further work27:39
Thank you28:32