def test_mandel_transform():
assert np.isclose(chaos.Mandelbrot_disp(0,0,10,50,100).mandel_transform(lambda x,c: x**4-x**2+c**2)[10,10], 1.2833333333333332)
"""
Creating the classic Mandelbrot set
======================================
"""
########################
# Some setup
# --------------
import chaoseverywhere as chaos
##################################
# Classic Mandelbrot
# --------------------------------
#
# First, let's compute the classic version of the mandelbrot set in black and white.
# It is well known that the Mandelbrot set is contained in the origin disk of radius 2 :math:`D((0,0),2)`, so we just iterate the formula :
#
# .. math::
#
# z_{n+1}=z_n^2+c,\ c\in\mathbb{C},
#
# with :math:`z_0=0\in\mathbb{C}`.
chaos.Mandelbrot_disp(0,0,2,t_max=150).disp_mandel()
def test_mandel():
test = chaos.Mandelbrot_disp(-.5, 0, 1,t_max=10,precision=20).mandelbrot()
assert test[0,0] == True
def test_mandel_loop():
test = chaos.Mandelbrot_disp(-.5, 0, 1).mandel_loop()[0,0]
assert np.isclose(test,1.5198773448773448)
# It is well known that the Mandelbrot set is contained in the origin disk of radius 2 :math:`D((0,0),2)`, so we just iterate the formula :
#
# .. math::
#
# z_{n+1}=z_n^2+c,\ c\in\mathbb{C},
#
# with :math:`z_0=0\in\mathbb{C}`. And the twist here is to change the value of the points whose modulus is beyond 2. Let's consider that we affect it with the value of the current iteration.
# For that and because it can be time consuming to calculate the modulus of numbers in an array (mainly because of the square root), we use the formula :
#
# .. math::
#
# \forall\ z \in\mathbb{C},\ |z|^2=z\bar{z}.
#
# And then, we don't compare it with 2, but :math:`2^2`.
mandel = chaos.Mandelbrot_disp(0, 0, 2, t_max=150).mandel_loop(go_up=False)
fig, ax = plt.subplots()
pict = ax.imshow(mandel, cmap='cool')
fig.colorbar(pict, extend='both')
plt.show()
########################################
# A little better...
# -------------------------
#
# The issue here is that chosing the iteration number creates jumps that may be not the smoothest and because :math:`z_0=0` and some points take a lot of iterations to make the sequence diverge (the one near the boundary), the constrast is not really visible.
# In order to correct that, we have a lot of choices, let's take :math:`\frac{1}{2+n}`, where :math:`n` is the current iteration of the sequence.
# We can also see that the Mandelbrot set is symmetric with respect to the real axis.
mandel = chaos.Mandelbrot_disp(0, 0, 2, t_max=150).mandel_loop(go_up=True)
fig, ax = plt.subplots()
import chaoseverywhere as chaos
import matplotlib.pyplot as plt
import matplotlib.animation as animation
#################################
# See a self-similar structure
# -----------------------------------
# The Mandelbrot set is clearly not a self-similar object. But inside it, we can see structures repeating themselves.
#
plt.figure()
plt.axis('off')
plt.imshow(chaos.Mandelbrot_disp(-1, -.3, 0.4-110/300,
t_max=100,
precision=400).mandelbrot(), cmap='bone')
plt.show()
######################################
# One way to animate a zoom
# -----------------------------------
#
# .. code-block:: python
#
# im_init = chaos.Mandelbrot_disp(-.5,0,1.5)
# im_init = im_init.mandelbrot()
# fig = plt.figure()
# im = plt.imshow(im_init, cmap='bone', animated=True)
""" ######################## # Some setup # -------------- import matplotlib.pyplot as plt import chaoseverywhere as chaos ########################### # Sparsity # -------------------- # # The use of projecting non-zeros values over an area to determine its area is very well known (it's even how most of us learn the Monte-Carlo algorithm to calculate an approximation of pi). # Let's say someone needs to do the same process with the Mandelbrot set. Then, a simple way to graphically overset the two objets is like below. fig = plt.figure() mandel = chaos.Mandelbrot_disp(-.5, 0, 1.5).mandel_loop(go_up=True) plt.imshow(mandel, cmap='Spectral') chaos.sparse_matrix(400, 400, .02) plt.show() #################### # Some values # ---------------------- # It can be estimated that the Mandelbrot set has an area between :math:`1.50` and :math:`1.51`. # It was proved by Mitsuhiro Shishikura that the Haussdorf dimension of the boundary of the Mandelbrot set equals :math:`2`. # # #
# Before we begin, a reminder of the 2D convolution between two matrix can be useful.
# In our case, we will use a :math:`3\times 3` kernel. So, the convolution of a kernel with a matrix is defined as the
# sum of the conter-row-wise by row-wise product of the elements ie the last element of the kernel is multiplied by the first of the matrix,
# the penultimate of the kernel (at the left of the last) is multiplied by the seconde one of the matrix (at the right of the first) and so
# on from the antepenultimate to the first one. In a formula we have :
#
# .. math::
#
# \begin{bmatrix} k_1 & k_2 & k_3 \\ k_4 & k_5 & k_6 \\ k_7 & k_8 & k_9 \\ \end{bmatrix} \star
# \begin{bmatrix} c_1 & c_2 & c_3 \\ c_4 & c_5 & c_6 \\ c_7 & c_8 & c_9 \\ \end{bmatrix} =
# k_1 c_9 + k_2c_8 + \dots + k_9c_1
#
# There are mutliple ways to get the edges of a shape using this method. We chose to use a kernel with only :math:`-1` on its borders
# and a value :math:`k_5=8=-\sum_{i=1,\ i\neq 5}^9 k_i`. We also need to pad the image in order to get all the pixels in it, and not
# forget the borders. Let's take a look at the result.
import chaoseverywhere as chaos
import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import convolve2d
mandel = chaos.Mandelbrot_disp(-.5, 0, 1.5).mandelbrot()
kernel_edge_detect = np.array([[-1, -1, -1], [-1, 8, -1], [-1, -1, -1]])
pad_mandel = np.pad(mandel, ((1, 1), (1, 1)), "maximum")
bound = convolve2d(pad_mandel, kernel_edge_detect,
mode='valid').astype(bool) * mandel
plt.imshow(bound, cmap='bone')
plt.axis('off')
plt.show()
import os
import sys
sys.path.append(
os.path.dirname(os.path.abspath(__file__)) + (os.path.sep + '..'))
import chaoseverywhere as chaos
chaos.Mandelbrot_disp(-.5, 0, 1, t_max=300).mandelbrot()
chaos.Mandelbrot_disp(-.5, 0, 1, t_max=300).mandel_loop()
chaos.bifurcation(show=False)