STRETCHING AND FOLDING DEMO
You might first want to review some of the definitions and terminology on chaos and attractors.
NOTE: This DEMO should help you get a better picture of how the concept of stretching and folding in a chaotic strange attractor works. But remember that is just a "hands-on analogy" of a very abstract process in phase space, that does not correspond to our "familiar" three dimensional space...
With these very abstract concepts in mind, lets go to our humble DEMO:
1) Secure the stick to the lab table using a clamp, or ask a partner to hold the meterstick.
2) Notice that the elastic band will stretch all the way to the other end of the stick. We will call X=0 the end of the stick where the band is attached, and X=1 the far end.
3) Notice that the screw is located approximately half-way in the stick. We will call X=1/2 its position (at the 50 cm mark).
4) Pick a point in the first half (0<X0<1/2) and mark it with pencil on the stick. We suggest X0 =0.1 (10 cm mark).
5) Stretch the band until its end reaches the screw (X=1/2) and copy with pencil the mark location into the band.
6) Now stretch the band all the way to the end of the stick (X=1).
7) After stretching, transfer the mark to the stick and record its value X1.
8) Repeat the process starting at X1. But when you get to a step such as Xm >1/2 the next step (to get Xm+1) is as follows:
-Use the last mark in the stretched band,
-Go around the screw and stretch the band until you reach X=0.
-Now transfer the mark on the band to the stick (to get: 1-Xm, you do not need to record this intermediate's step value)
-Stretch as in 5),6) 7) to get Xm+1 = 2 (1-Xm). Record the value of Xm+1.
9) Iterate 10 times the process ("rules" 5 through 8), using the last mark on the stick as the new starting point.
10) By construction, you should have 10 marks on the wood stick, all between 0 and 1. Notice its distribution.
11) Repeat this "game" using a different pencil's color to not get confused and starting a bit away (for instance X0' =0.11, or 11 cm mark) from the first game's starting point X0. You can think of initial conditions being 10% off between cases (or 10% "error").
12) What are your conclusions? Can you see how once very close points might become separated in the process, while distant points might become neighbors?
13) Pair up with next table, and agree on a common new third starting point. Go back to your table, iterate 10 times, and compare your ending point ("X10") with next table's X10. Do they agree? What do you conclude?
This non-linear iterated map we represented in this stretch and fold game is actually called the "bakers map" (imagine replacing the elastic band by dough being kneaded):
Xn+1 = 2 *Xn ..................if 0<Xn <1/2
Xn+1 = -2*Xn +2..............if 1/2<Xn<1
Fig No . 2 :
Questions to report:
a) List the 10 values (measure up to mm precision) of Xi for one of the trials.
b) Do a graphical iteration of the map that represents the sucession of 10 points Xi you obtained with the elastic band demo. You might find convenient to draw the Xn+1 = Xn diagonal line. Compare with a).
c) What are the fixed points of this map? Are they stable or unstable?
d) Do you notice a sensitive dependence of the iteration on the initial value Xo?
Extra Credit Questions:
e) How would you estimate the Liapunov exponent L for this process?
f) How would you estimate the "time horizon" after which subsequent predictions start to be practically impossible?
!A PROGRAM THAT CALCULTES THE GRAPHICAL ITERATION OF THE TENT MAP
! ITERATION OF TENT MAP EQUATION
! Xn+1=2Xn if Xn<=1/2
! Xn+1=2(1-Xn) if Xn>1/2
SET MODE "graphics"
SET WINDOW 0,1.333,-.1,1
BOX LINES 0,1,0,1
PLOT TEXT, AT 1,0.95: "(1,1)"
SET COLOR "red"
SET COLOR "black"
PLOT TEXT, AT 1.6,1.7: "y=x"
FOR x= 0 to 1 step 0.01
IF x<=0.5 THEN LET Xn= 2*x
If x>0.5 THEN LET Xn=2*(1-x)
SET COLOR "red"
GET MOUSE x,y,s
IF s=2 THEN EXIT DO
SET CURSOR 1,1
PRINT USING "x=#.####, y=#.####": x,y
IF j=0 THEN SET COLOR "black"
LET j = j+1
IF j=1 THEN SET COLOR "blue"
IF j=2 THEN SET COLOR "red"
IF j=3 THEN SET COLOR "brown"
IF j=4 THEN SET COLOR "green"
IF j>4 THEN SET COLOR "black"
set cursor j+1,1
PRINT "x0 is: ";x0
PLOT TEXT, AT x0,-.05 :"x0"
BOX DISK x0-0.01, x0+0.01,-0.01,0.01
FOR i=1 to 10
IF x0<=0.5 THEN LET X1= 2*x0
If x0>0.5 THEN LET X1=2*(1-x0)
BOX DISK x0-0.01, x0+0.01,x1-0.01,x1+0.01