Parabolic Julia set by ModAng method. Quadratic Mandelbrot by Gaussian method.


The Wave Function Method Viewer performs effective inverse iteration by repeated forward mapping of f(z) onto an eventual wave function of the complex coordinates. This means the values of the iterated range correspond to n-th pre-images of these range points in the original domain. I hit upon the method extending representations of complex function to iterations. So far as I know it is the only method that portrays the discrete chaotic dynamic in motion.

See: for full manual, downloads and further research.

Mac XCode Wave Function Method Viewer: Application - Source

Wait for the first sequence to complete, then insert c values for a Julia set or tick Mandelbrot of the parameter plane, and click redraw. The speed slider can be adjusted to slow the motion and the file menu can be used to stat and stop the motion. You can enter values into the text fields to control the display position, scale, Julia c values and number of iterations. You can also select several functions and the wave function method from the pop-down menus.

Herman ring by PolarWaveA, Riemann zeta Julia(0) by Gaussian.

There are five eventual wave functions. PolarWaveR uses sinusoidal waves of the absolute value and angle of the complex number with a lower frequency radial factor. PolarWaveA has a lower frequency angular factor. CartWaves uses two colour Cartesian waves. ModArg uses the modulus and argument in the same manner as DHViewer and RZViewer option 0. Gaussian uses a small bell-shaped e-z^2 surface over the origin. When this method completes, it gives in the last frame the maxima of all the previous iterations giving a view of Mandelbrot and Julia sets in terms of the points which are eventually fixed on 0. Doing this on the quadratic Mandelbrot set gives a succession of frames displaying locations of the bulbs and islands of every period, because the superattracting cycles of each period have zero derivative and hence iterate through 0. The other wave function methods also display these periodicities.

Several functions are included including z2+c, cz(1-z), z3-z+c (showing the effects of one critical value), cCos(z), the Herman ring function cz2(z-4)/(4z-1) and Riemann zeta.

The Riemann zeta function is included to a 40 term series Z(z)=Sum[1,n](n-z). Displaying the Julia set of the zeta function for c=0 in Gaussian mode gives a sequence of images of the zeta zeroes and their pre-images, as the zeros iterate in one step to 0 and their pre-images do in n steps. Use a scale of 40 to view.

The zeta Mandelbrot set is best viewed as a whole in ModArg as the Gaussian method displays only the fractal region neighbouring the singularity at z=1. This can be viewed with a scale of 4.

The application begins with parabolic quadratic example. The corresponding value for a Siegel disc for z2+c is c=-0.3905 - 0.5868i. The irrational flow of the Herman ring can be viewed in PolarWave mode with a scale of 6 origin (2,0) c=-0.7374 -0.6755i and PolarWaveA.