Bifurcation diagram for the logistic map n-> r n (n-1). Also called Feigenbaum diagram

logisticmap.py

import turtle as t

xstart = 3.2
ystart = 0
xend = 4
yend = 1
xstep = 0.5*(xend-xstart)/t.screensize()[0]


convergesteps = 100
plotsteps = 100

t.setworldcoordinates(xstart,ystart,xend,yend)
t.penup()
t.hideturtle()

def logistic(r,n):
    return r*n*(1-n)

def plotn(n,x,y=0.5):
    """
    plot n points.
    If the sequence has converged these will all be in the same
    place. Use x for the logistic parameter and y for initial value 
    """
    t.tracer(0)
    for i in range(n):
        t.setpos(x,y)
        t.dot(1)
        y = logistic(x,y)
    t.update()

def converge(x,y=0.5,loops=convergesteps):
    """
    loop the logistic map to converge to an attractor (orbit, or strange)
    return the final value of the sequence to use as an initial value for the
    ploting routine.

    The problem is the number of loops (100 by default)
    is both too big -> the sequence will often have converged before that point
    or too small -> at the interesting biurfication points it will take longer
    But how to decide when to stop converging and start plotting? For example
    just before the 2->4 biurification the sequence will almost enter a 4 loop,
    but in two pairs of close values, which become progressively closer
    [0, 0.1   1, 1.1 ; 0.02, 0.08 1.02 1.08 ...] eventually convegint to a 2
    -orbit. How is the 'puter to recognize this
    """
    for i in range(loops):
        y = logistic(x,y)
    return y

def main():
    x=xstart
    while x <= xend:
        y = converge(x)
        plotn(plotsteps,x,y=y)
        x += xstep

main()
t.exitonclick()