#
# program to plot a chaotic function, connecting dots with lines
#
import turtle               # allows us to use turtles

#

# these control the plotting and start of the function
#
startx = 0.26               # initial value of x for the chaotic function
scale = 200                 # need to scale to make the plot visible

#
# the particular chaotic function (see lecture 2)
def chaos(x):
    return 3.9 * x * (1 - x)

#
# set up the drawing
#
# window stuff first
wn = turtle.Screen()        # draw the window for the picture
wn.bgcolor("red")           # make the background color red
wn.title("Graph of a Chaotic Function") # put up a title
# turtle (or pen) stuff next
kelly = turtle.Turtle()     # create a turtle, called "kelly"
kelly.shape("classic")      # just an arrow
kelly.color("white")        # white contrasts nicely with red

#
# draw the axes
#
# first, the x axis, from 0 to 1 (scaled)
kelly.penup()
kelly.setpos(0, 0)
kelly.pendown()
kelly.goto(1*scale, 0)
# now the y axis, from 0 to 1 (scaled)
kelly.penup()
kelly.setpos(0, 0)
kelly.pendown()
kelly.goto(0, 1*scale)
kelly.penup()

#
# mark the axes
#
# first, make a mark on the y-axis where y = 1
kelly.goto(-0.01*scale, 1*scale)
kelly.pendown()
kelly.goto(0.01*scale, 1*scale)
kelly.penup()
# now add 0 and 1 
kelly.goto(-0.01*scale, 0.98*scale)
kelly.write("1", False, "right")
kelly.goto(-0.01*scale, -0.02*scale)
kelly.write("0", False, "right")
# and label the axis
kelly.setpos(-0.01*scale, 0.48*scale)
kelly.write("f(x)", False, "right", ("Arial", 10, "normal"))
kelly.setpos(0.5*scale, -0.1*scale)
kelly.write("time", False, "center", ("Arial", 10, "normal"))
# now write something below the y-axis
kelly.goto(0.5*scale, -0.2*scale)
kelly.write("Graph of f(x) = 3.9 * x * (1-x) from x = 0.26", False, "center", ("Arial", 12, "normal"))

#
# move to the first point, which has x = 0
#
x = startx
kelly.setpos(0, chaos(x))
# start the line there
kelly.pendown()
for i in range(1, 200):
    x = chaos(x)
    # move to the position; this draws the line (pen is down)
    kelly.goto(i, x * scale)



#
# now hide the turtle and just sit there, displaying the result
#
kelly.hideturtle()          # make the turtle invisible
wn.mainloop()               # wait until the user closes the window