Bifurcation Diagram

Generating the famous fractal, which can be used to model populations with various cycles, generate pseudo-random numbers and determine one of nature’s fundamental constants, the Feigenbaum constant \delta.
The fractal nature comes from iteratively applying a simple function, f(x) = \lambda \cdot x \cdot (1-x) with 0 \leq \lambda \leq 4, and looking at its poles.
The resulting image looks mundane at first, when looking at 0 \leq \lambda \leq 3, though the last quarter section is where the interesting things are happening (hence the image below only shows the diagram for 2 \leq \lambda \leq 4).
From \lambda = 3 on, the diagram bifurcates, always doubling its number of poles, until it enters the beautiful realm of chaos and fractals.

Bifurcation Diagram lambda in range [2; 4]
For more on bifurcation, fractals and \delta, I refer to this Wikipedia entry and WolframMathworld.


# Python 2.7.7 Code
# Jonathan Frech, 24th of March 2017

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Mandelbrot Set II

Over a year ago, I published my first Mandelbrot Set viewer, a Python program using pygame. Since then, I have published a rather short program highlighting errors that can occur when calculating the set (Mandelbrot Set Miscalculations).
Since my first viewer was in Python, which is an interpreted programming language, and I wanted to make my viewer faster, I decided to write one in Java. I was hoping for a speed increase since Java is compiled and thus should run at higher speeds. I was not disappointed. The new Java-based viewer runs noticeably faster and additionally I added a lot of new features, all listed below.

Controls

  • Left-clicking and dragging draws a zoom frame, single left-clicking removes the frame
  • Right clicking (and optionally dragging) moves the zoom frame
  • Space zooms into the zoom frame
  • F1 moves one step back the zoom history
  • F2 shows the path a complex number at the cursor’s position follows when the function is iteratively applied
  • F3 shows the \mathbb{R} and \mathbb{R}i axis
  • F4 displays the current cursor’s corresponding complex number
  • F5 toggles between showing and hiding the menu (text in the left upper corner describing the viewer’s functions and current states)
  • F6 increments the exponent (going from f_c(z)=z^2+c to f_c(z)=z^5+c in whole-number steps)
  • F7 toggles between the Mandelbrot set and the filled Julia set
  • F8 toggles between previewing a small filled Julia set at the cursor’s position based upon the cursor’s complex number
  • F9 completely resets the zoom and zoom history
  • F11 (or F) toggles between fullscreen and windowed mode
  • F12 quits the application
  • L increases the color depth (starting at 256 and increasing in steps of 256)
  • Q saves the current image to disk

To use this application, you can either trust me and download the .jar-file or view the source code listed below, verify it and compile the program yourself.
The program will start in fullscreen mode, to change to windowed mode, just press F11 (as listed above).

The standard Mandelbrot setA Filled Julia setA Mandelbrot set using the fourth powerA deeper zoom into a fourth power Mandelbrot setA filled Julia set of the fourth power


// Java Code
// Jonathan Frech 14th of September, 2016
//         edited 15th of September, 2016
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//         edited 14th of December , 2016
//         edited 13th of January  , 2017

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Mandelbrot Set Miscalculations

While developing a Java program to create an image of the Mandelbrot set, I stumbled upon a small error which completely changes the set’s look. To fix this bug, you need to swap two lines of code.

The bug arises when trying to convert convenient Python features to Java.
To iteratively apply the function z \to z^2 + c, you update your complex number z a certain amount of times. When you are not using a complex number class, but instead you use two floating point numbers (in Java doubles to gain precision) \texttt{a} and \texttt{b} to define the real and imaginary part (z = \texttt{a} + \texttt{b} \cdot i), logically both numbers need to be updated.
In Python you may write the following, when c is defined as being a complex number with parts \texttt{c} and \texttt{d} (c = \texttt{c} + \texttt{d}).

a, b = a**2 - b**2 + c, 2 * a * b + d

Which seems to be very similar to those two lines.

a = a**2 - b**2 + c
b = 2 * a * b + d

But notice that in the first code snippet you define a tuple consisting of the real and imaginary part and then assign it to the variables. The first snippet really looks like this.

t = (a**2 - b**2 + c, 2 * a * b + d)
a, b = t

Using this assignment of variables, which corresponds to z \to z^2 + c, you get an image of the correct Mandelbrot set.
mandel_correct
In contrary, the second code snippet assigns the old \texttt{a} its new value, then uses this new \texttt{a} to define the value of new \texttt{b}, thus does not calculate z \to z^2+c, which is equivalent to z \to (\texttt{a}^2-\texttt{b}^2+\texttt{c}) + (2\cdot\texttt{a}\cdot\texttt{b}+\texttt{d}) \cdot i, but rather z \to (\texttt{a}^2-\texttt{b}^2+\texttt{c}) + (2 \cdot \texttt{a}^2 \cdot \texttt{b}-2\cdot \texttt{b}^3+2\cdot\texttt{b}\cdot\texttt{c}+\texttt{d})\cdot i.

In Java it would look like this.

a = a*a - b*b + c;
b = 2 * a * b + d;

Which results in this rather unusual depiction of the famous fractal.
mandel_miscalculation
You can easily avoid this bug when using two sets of variables to define old z and new z, as shown in the following.

_a = a*a - b*b + c;
_b = 2 * a * b + d;
a = _a;
b = _b;

Or you can define variables \texttt{asqr} = \texttt{a}^2 and \texttt{bsqr} = \texttt{b}^2 and swap the assignment. Using variables for the squares of the parts of z also helps to improve performance.

b = 2 * a * b + d;
a = asqr - bsqr + c;
asqr = a*a;
bsqr = b*b;

// Java 1.8 Code
// Jonathan Frech, September / November / December 2016

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