Intro To Computer Science Lesson Plan : Visualizing Chaos via Orbit Diagrams of Logistic Map and the Henon Map

Last Fall I taught a class to incoming Freshman to introduce "interesting" ideas from the field of Computer Science. I tried to create some labs that would engage the students without being too technical or too dumbed-down. One successful lab had the students take a well-known chaotic function known as the logistic equation x(n+1) = r*x(n)*(1-x(n)) and asked them to try to form a visualization. Here are some of the results of the students.

Here are project results from CS0 students: Daniel Hagerstrand, Cody Mays, and Scott Resnick

Here is basic python code that generates the graphic for the orbit diagram of the logistic equation:

from Tkinter import *

import random

root = Tk()

canvas = Canvas(width=1000, height=750, bg='white')

canvas.pack(expand=YES, fill=BOTH)

text = canvas.create_text(50,10, text="Chaos Test")



def drawcircle(canv,x,y,radius,color):

    canvas.create_oval(x-radius,y-radius,x+radius, y+radius,width=0,fill=color)



# Feigenbaum bifurcation values.

# b1 = 3, b2 = 3.449490…,

# b3 = 3.544090…, b4 = 3.556441…, b5 = 3.568759…, and b6 = 3.569692



r=2.4

while r < 4:

    for pix in range(50):

        x = random.random()

        for i in range(100):

            x = r*x*(1-x)

            drawcircle(canvas,x*1000,(r-2.4)*(750/1.6),1,'blue')

    r+=0.005

root.mainloop()

Here is basic python code that generates the graphic for the orbit diagram of the Henon Map:

from Tkinter import *

import random

root = Tk()



def drawcircle(canv1,x,y,radius,color):

    canv1.create_oval(x-radius,y-radius,x+radius, y+radius,width=0,fill=color)



canvas = Canvas(width=1400, height=800, bg='white')

canvas.pack(expand=YES, fill=BOTH)

text = canvas.create_text(100,10, text="    Orbit Diagram of the Henon Map")





def HenonMap(a,b,x,y):

	return y + 1.0 - a *x*x, b * x



# Map dependent parameters

a = 1.4

b = 0.3

iterates = 100000



xtemp = 0.1

ytemp = 0.2



for pix in xrange(0,iterates):

    xtemp, ytemp = HenonMap(a,b,xtemp,ytemp)

    drawcircle(canvas,400*xtemp+750,1000*ytemp+400,1,'blue')



root.mainloop()