# LogisticCobweb.py:
#	Plot the Logistic Map's quadratic function
#   And show iterating from an initial condition

# Import modules
from numpy import *
# Plotting
from pylab import *

# Define the Logistic map's function 
def LogisticMap(r,x):
	return r * x * (1.0 - x)

# Setup x array
# Note how we make sure x = 1.0 is included
x = arange(0.0,1.01,0.01)
# We set r = 3.678, where two bands merge to one
r = 3.678

# Setup the plot
# It's numbered 1 and is 6 x 6 inches, to make the plot square.
# On my display this didn't turn out to be square, so I've fudged.
figure(1,(6,5.8))
# Note how we turn the parameter value into a string
#   using the string formating commands.
TitleString = 'Logistic map: f(x) = %g x (1 - x)' % r
title(TitleString)
xlabel('X(n)')   # set x-axis label
ylabel('X(n+1)') # set y-axis label

# We plot the identity y = x for reference
plot(x, x, 'b--', antialiased=True)

# Here's the Logistic Map itself
plot(x, LogisticMap(r,x), 'g', antialiased=True)

# ... and its second iterate
# Set the initial condition
state = 0.2
# Establish the arrays to hold the line end points
x0 = [ ] # The x_n value
x1 = [ ] # The next value x_n+1
# Iterate for a few time steps
nIterates = 20
# Plot lines, showing how the iteration is reflected off of the identity
for n in xrange(nIterates):
	x0.append( state )
	x1.append( state )
	x0.append( state )
	state = LogisticMap(r,state)
	x1.append( state )

# Plot the lines all at once
plot(x0, x1, 'r', antialiased=True)

# Use this to save figure as a bitmap png file
#savefig('LogisticCobweb', dpi=600)

# Display plot in window
show()