The Mandelbrot set¶

The creation¶

The Mandelbrot set \(\mathcal{M}\) is defined as :

\[\mathcal{M}=\left\{ c\in\mathbb{C},\ (z_n)_n \text{ is bounded},\ \text{with } z_{n+1}=z_n^2+c,\ z_0=0\right\}.\]

Given the fact that it is a set of points, we can compute it using a class Python object. Thus, determine which arguments to pass in order to instantiate one set, and what methods to compute. What a user (you hopefully) might want to ask oneself is:

where should I center the picture of the Mandelbrot set?
do I want the whole set or just a special part? If the latest, what square exactly would I want to look at?
do I need of lot of iterations of the sequence: meaning more time to make, or can I be satisfied with a reasonable number of iterations?

Note

For only aesthetics purposes, we chose not to let the user display a rectangular part of the Mandelbrot set but only a square one.

So, the windows that will be displayed will be a square centered at the point \(a+bi\). The user will input this point separating the real part and the imaginary part in the two arguments x and y. The size of the square is determined by the facteur argument representing the half-length of the side of the square one’d like to display.

A summary to keep in mind¶

Methods	Output
	numpy array	saved video	matplotlib plot
anim_puiss_mandel		X
anim_pics_mandel		X
animate_mandel_plt		X
disp_mandel			X
mandel_loop	X
mandel_transform	X
mandelbrot	X

The class methods¶

class Mandelbrot_disp(x, y, facteur, t_max=100, precision=400)¶

Creates the mandlebrot graph and zooms in on this graph.

Moreover, there are functions who display the Mandlebrot set in 2D and 3D with animated video.

Parameters

x (float) – coordinate of the image’s center
y (float) – coordinate on the Imaginary axis
facteur (float) – the remoteness of the image’s center
t_max (integer) – the iteration’s number of the sequence (zn)
precision (integer) – number of terms in the array (using to build the image)

anim_pics_mandel(go_up=True, puiss=2)¶

Shows the Mandlebrot set in 3D. It saves the video in .avi.

Parameters

go_up (boolean) – type of display, using in mandle_loop function
puiss (integer) – the exponent in the Mandlebrot equation, using in the mandle_loop function

anim_puiss_mandel(remove=True)¶

Animates other powers in the Mandeblrot equation.

As the main equation of the Mandelbrot uses a power of 2, we display here the Mandelbrot set for all the integers between 2 and a hundred.

Returns: successive zoom of the Mandlebrot set
Return type: Animation / video

animate_mandel_plt(x=- 1, y=- 0.3)¶

Zoom in on the Mandlebrot graph.

This animation zooms in on the point (x,y) to see the fractals.

Parameters

x (float) – coordinate on the real axis of the point to zoom in.
y (float) – coordinate on the imaginary axis of the point to zoom in.

Returns

the animation of the Mandlebrot’s zoom saved in .avi

Return type

matplotlib plot

disp_mandel()¶

Using the array of the mandlebrot function, this function shows the mandlebrot set.

Returns: plot the Mandlebrot set
Return type: matplotlib plot

mandel_loop(go_up=True, puiss=2)¶

Determines coordinates of points, in the Mandlebrot set and the speed of divergence.

Parameters

go_up – type of display, condition to give values in mandel’s array
puiss – the exponant in the Mandlebrot equation

Type

boolean

Returns

the Mandlebrot set in colors

Return type

numpy array

mandel_transform(FUN)¶

Determines coordinates of points in the transformation of the Mandelbrot set.

Parameters: FUN (function) – function that replaces the mandelbrot recursive equation
Returns: the Mandlebrot set transformed
Return type: numpy array

mandelbrot()¶

Gives a boolean matrix reprensenting the Mandelbrot set.

We affect the value True if the point is in the Mandlebrot set, false otherwise.

Returns: a matrix indicating whether or not the point is in Mandlebrot set.
Return type: numpy array

Additional function in the submodule¶

As part of the link with the logistic map, we needed a function that could return a stairs-like data about the different values that would take the map \(z_{n+1}=z_n^2+c\) with different values for \(c\). See the bifurcation video in the gallery for the actual plot.

mandel_branch_points(x0, mu, nb_iter=20)¶

Fives the coordinates of Mandlebrot points.

Starting with the coordinate of (x0,0), the function applies the Mandlebrot sequence. Each coordinates is appended in a list, useful to draw stairs of the recursive sequence.

Parameters

x0 (float) – the starting point included in [0,1]
mu (float) – the common ratio of the Mandlebrot sequence, included in [0,1]
nb_iter (integer) – the number of coordinates put in our final list

Returns

list of Mandlebrot points coordinates

Return type

list of tuples

One can use this function to produce the same plot in the top-right of the video in the bifurcation diagram section of the gallery using the code below.

plt.style.use(['ggplot', 'dark_background'])

data = zip(*chaos.mandel_branch_points(0,-1,50))
ax2 = plt.subplot()
x = np.linspace(-2, 1, 400)

line, = ax2.plot([],[], color='red', alpha=1, lw=4)
line.set_data(data)

courbe, = ax2.plot([], [], color='dodgerblue', alpha=1, lw=2)
courbe.set_data(x, x**2-1)
ax2.plot(x, x, color='orange')
plt.show()