Hello,

I recently came across this forum, and after seeing the wonderful

Bifurcation Fractal Plotter by nkercso, it gave me the idea to share a little program I did some years ago, and just recently improved.

I’m not sure where this topic shall be published between

**Programming**,

**Fractal Programs** or

**Fractal Math, Chaos Theory & Research**… So, to the mods, feel free to move this topic to the correct area. For the moment, I assume that’s the background theory which is more important to discuss, the program being here to illustrate… I actually never read anything related to those curves, and when I got the idea about plotting them, that was the main motivation for writing this program. Please let me know if you have any link related to these curves, fractals (are they?).

The idea of the program was to plot kind of bifurcation fractals based on the Mandelbrot calculation. First draw the Mandelbrot set, and then draw a line onto it. We will follow that line, and plot for each value of this axis, the calculation of the madelbrot fractal, but plotted like a bifurcation fractal.

For example, let's start from A (=a+bi) and C (=c+di), for real k=[0…1], I’ll consider K=(1-k)*A + k*B (=((1-k)*a+k*c) + ((1-k)*b+k*d)i), and I’ll compute the Mandelbrot calculation.. f(z) = z^2+k, and for each k plot the values of module and the argument of the complex numbers z; like we would do with the logistic equation. Of course we can enlarge the graphic for k < 0 and k > 1 (like we extended the logistic equation for negatives values)

Here is some screen shoots of the calculations to show…

*pick a line over the Mandel set**the bifurcation diagram following that axis**a zoom of a part of it**https://lut.im/razCijR9Bv/hjxW87012GuxF3Bt.jpg**with more iterations**the bifurcation diagram shows the module in the positive area, and behind it the argument (from Pi-4 to –Pi-4).**here is another example with zoom over the phase, you can see this is circular of course**Here is another example again**the bifurcation diagram (with default zoom values)**zooming out: modules and arguments**zooming in some details**with more iterations**plotting a Julia**the same algorithm can be applied…*The program is written in Java (you need a JVM to be installed) and is far to be optimized… I guess the computations can be optimized… The GUI is not really finished… and I’m not very good at graphics programming. I’ve recently optimized the plotting using the algorithm used by nkercso’s program… Also the default zoom values are not optimized, but this is no big deal…

To use the program, you can select Julia to click on the figure and plot the Julia for the point.

You can select zoom and click to zoom-in, or right-click to zoom out. You can also manually set zoom value (sorry the GIU forum is pretty poor).

And then select Bifurcation to create a bifurcation from a drawed line.

Thanks for reading. Thanks for trying !