In [1]:
# Code from last time
# Code to draw colored Julia set
import numpy
def draw_julia_color(c, max_iter, xmin, xmax, ymin, ymax):
"""Draws colored Julia set for z^2+c.
Color is hotter for points that take longer to escape.
Should only use with |c|<2.
max_iter is max number of iterations
xmin, xmax smallest, largest x-coordinates to check."""
height = 600 # height in pixels of image
width = 600
max_abs=2 # if |z|>2 then the iterates f(z),f(f(z)),... escape to infinity
xinc=(xmax-xmin)/width # width of each pixel
yinc=(ymax-ymin)/height
julia = numpy.zeros((height,width)) # initialize a numpy array of all zeros
for x in range(width):
for y in range(height):
z = complex((xmin + xinc*x),(ymin + yinc*y))
iters = 0 # count of number of iterations
while iters<max_iter and abs(z)<max_abs:
z = z*z + c
iters += 1
# if don't escape, iters = max_iter
# if it does escape, iters is lower the faster it escapes
julia[y,x] = iters/max_iter
return matrix_plot(julia, cmap='hot', origin="lower") #'hot' is a particular color profile
In [2]:
# f_0 (z) = z^2 + 0
draw_julia_color(0, 20, -1, 1, -1, 1)
# Julia set is the boundary of the white region, the unit circle
Out[2]:
Definition of Mandlebrot set
Let $f_c(z) = z^2 + c$ (these give the "quadratic family"). The Mandlebrot set is defined as $$M = \left\{c : \lim_{n\to\infty} f_c^{\circ n}(0) \ne \infty \right\},$$ i.e. those $c$ for which the orbit of $0$ under iteration of $f_c$ does not escape.
In [3]:
# Code to draw the Mandlebrot set
def mandlebrot(max_iter, xmin, xmax, ymin, ymax):
"""Draw Mandlebrot set. Each point c
corresponds to function f_c(z)=z^2 + c.
Assuming |c|<2. (this doesn't really matter)"""
height = 400 # make lower res, so it will be faster
width = 400
max_abs=2
xinc=(xmax-xmin)/width # width of each pixel
yinc=(ymax-ymin)/height
mand = numpy.zeros((height,width)) # grid represent mand set
for x in range(width):
for y in range(height):
c = complex((xmin + xinc*x),(ymin + yinc*y))
iters = 0
z = 0 # start with z=0, then will apply f_c repeatedly
while iters<max_iter and abs(z)<max_abs:
z = z*z + c
iters += 1
# if doesn't escape, iters = max_iter
# if does escape, iters is small if it escapes quickly
mand[y,x] = iters/max_iter
return matrix_plot(mand, cmap='hot', origin="lower")
In [5]:
mandlebrot(30, -2,2, -2,2)
Out[5]:
In [6]:
# Feigenbaum point c = -1.401155
# z^2 + c
p = -1.401155 # will zoom in centered here
s = 0.1 # amount to zoom in
mandlebrot(100, p-s,p+s, -s,s)
Out[6]:
In [8]:
# Feigenbaum c = -1.401155
# z^2 + c
p = -1.401155
s = 0.02
mandlebrot(200, p-s,p+s, -s,s)
# this picture looks almost the same as above,
# even though it zoomed in
# there are smaller and smaller bulbs
# accumulating to the point p
Out[8]:
In [26]:
# Draw julia sets corresponding to pts in mandlebrot set
# Julia set for f_c(z) = z^2 + c, c=-1.401155
c=-1.401155
draw_julia_color(c, 100, -2, 2, -2, 2)
# Julia set is connected
Out[26]:
In [30]:
# trying a value not in mandlebrot set
# close to feigenbaum (a bit above)
draw_julia_color(complex(c,0.1), 400, -1, 1, -1, 1)
# get disconnected Julia set
Out[30]:
In [33]:
# trying a value in mandlebrot set
# close to feigenbaum
draw_julia_color(complex(c+0.1,0), 400, -1, 1, -1, 1)
# is connected
Out[33]:
In [34]:
# zoom in on previous
draw_julia_color(complex(c+0.1,0), 400, 0.25-0.2, 0.25+0.2, -0.2, 0.2)
Out[34]:
