Picture Gallery

Some chaos pictures:

Newton's method in the complex plane: Newton's method is an iterative method to find the zeros of a complex polynomial, here the third roots of unity. Rather surprisingly, the basins of attraction for the three roots are fractal: which root we find depends on the starting point, and in some regions of the complex plane, an infinitesimal displacement of the starting point changes the attracting root. The picture shows a part of the complex plane, and every point is used successively as a starting point. It is then colored by the color of the root it ended up in, intensity corresponding to the number of iterations necessary to reach it.

Logistic map: The logistic map is defined for all x between 0 and 1 and r between 0 and 4 by x(t+1) = r*x(t)*(1-x(t)). For small r, it has a stable fixed point: iterating the map, for long t it will finish at a unique value. For larger values of r, bifurcations occur: x oscillates between 2, then 4, then 8 ... values, and finally we have chaotic behavior, with small regular stripes embedded. In this picture, the horizontal axis is r, and the vertical is x, and I have plotted several hundreds of iterations, skipping an initial transient.

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