Verhulst Mandelbrot#
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verhulst mandelbrot#
Most examples work across multiple plotting backends, this example is also available for:
Example showing how bifurcation diagram for the logistic map relates to the Mandelbrot set according to a linear transformation. Inspired by this illustration on Wikipedia.
from itertools import islice
import numpy as np
import holoviews as hv
from holoviews import opts
hv.extension('bokeh')
Defining Mandelbrot and Logistic Map#
# Area of the complex plane
bounds = (-2,-1.4,0.8,1.4)
# Growth rates used in the logistic map
growth_rates = np.linspace(0.9, 4, 1000)
# Bifurcation points
bifurcations = [1, 3, 3.4494, 3.5440, 3.5644, 3.7381, 3.7510, 3.8284, 3.8481]
def mandelbrot_generator(h,w, maxit, bounds=bounds):
"Generator that yields the mandlebrot set."
(l,b,r,t) = bounds
y,x = np.ogrid[b:t : h*1j, l:r:w*1j]
c = x+y*1j
z = c
divtime = maxit + np.zeros(z.shape, dtype=int)
for i in range(maxit):
z = z**2 + c
diverge = z*np.conj(z) > 2**2
div_now = diverge & (divtime==maxit)
divtime[div_now] = i
z[diverge] = 2
yield divtime
def mandelbrot(h,w, n, maxit):
"Returns the mandelbrot set computed to maxit"
iterable = mandelbrot_generator(h,w, maxit)
return next(islice(iterable, n, None))
def mapping(r):
"Linear mapping applied to the logistic bifurcation diagram"
return (r /2.0) * ( 1 - (r/2.0))
def logistic_map(gens=20, init=0.5, growth=0.5):
population = [init]
for gen in range(gens-1):
current = population[gen]
population.append(current * growth * (1 - current))
return population
Plot#
bifurcation_diagram = hv.Points([(mapping(rate), pop) for rate in growth_rates for
(gen, pop) in enumerate(logistic_map(gens=110, growth=rate))
if gen>=100]) # Discard the first 100 generations to view attractors more easily
vlines = hv.Overlay([hv.Curve([(mapping(pos),0), ((mapping(pos),1.4))]) for pos in bifurcations])
overlay = (hv.Image(mandelbrot(800,800, 45, 46).copy(), bounds=(-2, -1.4, 0.8, 1.4))
* bifurcation_diagram * hv.HLine(0) * vlines)
hv.output(size=150)
overlay.opts(
opts.HLine(color='k', line_dash='dashed'),
opts.Image(cmap='Reds', logz=True, xaxis=None, yaxis=None),
opts.Points(size=0.5, color='g'),
opts.Curve(color='teal', line_width=1))
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