FracPy

How to use

Clone this repository and install sympy, numpy, matplotlib, and numba.

To explore the Mandelbrot and Julia sets given by the family $f(z) = z^2 + C$ just run:

python ./fracpy.py

To explore the plots of other functions or 1-parameter families of functions, you need to open the python interpreter or a jupyter notebook and run the following. First, you have to import fracpy's main class DSystem and some symbols from sympy:

>>> from fracpy import DSystem  # class to store dynamical systems
>>> from sympy.abc import z, c  # z and c are now sympy symbols and can be used in expressions

Then you have to create a dynamical system using those variables and then call its view method.

>>> expr = z**2 - c * z
>>> pol = DSystem(z, expr, crit=c / 2)
>>> pol.view(mandel_center=-1.0, mandel_diam=8.0)

DSystem takes as arguments the function variable, the expression that determines the functions (it can have at most one parameter), and a critical value to plot the bifurcation locus (that can depend on the parameter).

More examples and details can be found on the examples.ipynb file.

REMINDER: sympy uses python conventions for math expressions, so:

• ** is the notation for exponentiation not ^;
• All multiplications must be explicit, e.g 2(1+z) is not valid, but 2*(1+z) is.

Also, you can import some basic functions and some math constants via sympy, for example

from sympy import E, I, pi, GoldenRatio, sin, cos, sqrt

imports the euler constant, the imaginary unit, $\pi$, the golden ratio, sine, cosine, and the square root.

Shortcuts

• z + <LeftClick>: Zooms in on the plot
• x + <LeftClick>: Zooms out on the plot
• s + <LeftClick>: Centers on pointer coordinates
• c + <LeftClick>: Chooses the parameter (only on Mandelbrot plot)
• t + <LeftClick>: Draws orbit starting at pointer coordinates (only on Julia plot)
• r: Resets view
• d: Hides orbit
• 1: Color only escaping orbits
• 2: Color bounded orbits by period
• 3: Color bounded orbits using $|z_n - z_{2n}|$
• 4: Color bounded orbits by preperiod
• 5: Color bounded orbits by absolute value of the derivative at attracting cycle
• 6: Color bounded orbits by argument of the derivative at attracting cycle
• <LeftArrow> and <RightArrow>: Shift color gradient

All the other settings can be changed by writing on the entries below the plot and pressing <Enter>.

Some Images

Rabbit:

Julia sets of the family $C\cos(z)$: