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Dynamical Systems
Daniel Waggoner
Department of Mathematics
Agnes Scott College
Decatur, Georgia 30030

This program can be used in the study of the dynamics of Newton's method for polynomials of degree less than sixteen. One can specify the polynomial by either entering its coefficients or entering its roots. The roots can be entered with the use of a mouse. The basins of attraction of each of the roots are drawn.

This program can also be used to study the parameter space associated with Newton's method for cubics, in which two roots are fixed at -1 and 1, and each pixel w represents the third root. Newton's method for the resulting polynomial

p(z) = (z-w)(z-1)(z+1)
is started at the center of the triangle formed by the three roots and the pixel w is colored according to which root that orbit converges to, or is colored black if convergence cannot be determined after a specified iteration count.

There is also a module for the study of the Mandelbrot set along with the associated Julia sets. After the Mandelbrot set is drawn it is possible to choose a point with a mouse and either draw its associated Julia set or draw the orbit of the point. These drawings are made in smaller windows which can be kept on the screen so that one can compare different points.

Download dy-syst.zip [109 KB].

Look at the readme file from the program.