# Animated COBWEB diagram of the Logistic Map
# written by:  Joseph D. Skufca, US Naval Academy

# Email: skufca@nadn.navy.mil
# This program generates an animation to graphically show how a small
# interval, under repeated application of the logistic map (4x(1-x))
# grows to cover the entire interval from 0 to 1, providing a graphical
# understanding of sensitive dependence on initial conditions under a
# chaotic map.

# The logistic map - f(x)=a*x*(1-x) was originally developed as a basic
# model of growth of a species when there is a limit to growth (as
# opposed to exponential growth, like bacteria).  f(x) predicts the
# population at the next time increment if the present population is x.
#  This simple model has a parameter "a."  If the parameter value is
# chosen to be a=4, this map shows the classic behavior of a chaotic
# system.  This map is studied  in nearly all initial courses in chaos
# theory.  The mathematics are very easy and can almost always be
# solved in closed form.

# As a chaotic map (unrelated to its use as a model), this map is well
# studied.  One of the interesting aspects is that the orbit (sequence
# of values) generated by the map is sensitive dependent on the initial
# value.  Although each value is calculated deterministically from the
# previous one, two orbits that start very close will eventually become
# very far apart.

# My animation shows what happens to initial values in the range from
# 0.4 to 0.41.  Over just eleven iterations of the map, the output
# could be anywhere between 0 and 1.  A classic approach to study of
# maps is to use a cobweb diagram, where the next value is determined
# graphically.  My animation, although calculated in closed form, uses
# a cobweb diagram to illustrate how the interval (0.4, 0.41)
# eventually maps to the interval (0,1).  Cobwebbing is an approach
# that is well described in other texts and would be familiar to anyone
# who has had a basic course in studying maps.

# Keywords: logistic, animation, sensitive dependence, population
# growth, cobwebbing

# The initial interval can be changed, as well as controlling the
# number of iterations animated.  The basic function for this animation
# uses the logistic function 4x(1-x).  Other maps can be animated using
# this program by changing the appropriate line that defines f(x).  If
# f(x) is altered, the program may fail due to the inability of the
# "maximize" and "minimize" algorithms to find an answer.  

> restart:
> with(plots):
> with(plottools):
> n:=11: # This specifies the number of iterations
> b:=array(1..2*n+1):
> a:=array(1..2*n+1):
> c:=array(1..n+1):
> x[1]:=.4:# This is the lower bound of the initial interval
> xd[1]:=.41:# This is the upper bound of the initial interval
> f:=x->4*x*(1-x):# This defines the map to use for iteration
> xintl:=0:xintu:=1:# This defines the domain of the map
> f1:=plot([f(x),x],x=xintl..xintu):
> c[1]:=plot(f(x),x=x[1]..xd[1],filled=true,color=gray):
> fplot:=display(f1,c[1]):
> i:=0:for i from 1 by 1 to n do  
> g:=minimize(f(x),x=x[i]..xd[i],location):xlow:=g[1]:xlowco:=rhs(op(1,o
> p(1,op(1,g[2])))):g:=maximize(f(x),x=x[i]..xd[i],location): 
> xhigh:=g[1]:xhighco:=rhs(op(1,op(1,op(1,g[2])))):x[i+1]:=xlow:xd[i+1]:
> =xhigh:c[i+1]:=plot(f(x),x=x[i+1]..xd[i+1],filled=true,color=gray): 
> a[2*i-1]:=line([xlowco,xlow],
> [xlow,xlow],color=blue):a[2*i]:=line([xhighco,xhigh],
> [xhigh,xhigh],color=blue)  od:

> b[1]:=fplot:
> i:=0:for i from 2 by 2 to 2*n do 
> b[i]:=display(f1,c[i/2],a[i],a[i-1]):b[i+1]:=display(f1,a[i],a[i-1],c[
> 1+i/2])  od:
> anim:=seq(b[i],i=1..2*n+1):
> display(anim,insequence=true);