## Chaos

- XLogo animation

This shows how a simple function can exhibit stable, cyclic and chaotic behaviour. The function used is t = a sin t, (in radians). The initial value of a is 3.8 and t is 0.2. The next value of t is then calculated as 3.8 * Sin 0.2. This value is plotted on a graph, then fed back into the function to obtain the next value of t.
The cursor up and down keys allow you to alter the value of a as the successive values of t are plotted. These will be stable, cyclic or chaotic at certain values of a.

To New
# set default screen, pen and turtle values
ResetAll SetScreenSize [400 400] HideTurtle
SetSC Black SetPC Green SetPS 1 PenUp
End

To Init
Make "A 3.8 Make "T 0.2     # starting values
Make "Delay 10     # reduce to speed up
Message [Use the Up and Down cursor keys to adjust the value of 'a']
End

To Title
SetPC White
SetPos [-160 180] Label [Plot of t= a Sin t where 1<a<5]
SetPos [130 180] Label [a= 3.8]
SetPos [-184 -186] For [W -5 5] [Label :W Forward 34]
SetPC Gray
SetPos [-161 -182] PenDown Repeat 4 [Forward 342 Right 90]
SetH 90 For [Y -148 124 34] [
PenUp SetPos (List Minus 159 :Y)
Repeat 85 [PenDown Forward 2 PenUp Forward 2]]
End

To Chaos
SetH 0
While ["True] [Make "X Minus 160
Repeat 86 [Make "T :A*Sin (:T*180/Pi)
PenUp SetPos (List :X Minus 12) PenDown
SetPC Red Forward :T*34 Wait :Delay Refresh
SetPC Green Back :T*34
Make "XX :X+40
If :XX> 180 [Make "XX :XX-344]
PenUp SetPos (List :XX Minus 181) PenDown
SetPC Black Forward 340
KeyIP Make "X :X +4]]
End

To KeyIP
If Key? [
If :Char=-38 [Make "A :A+0.1 Make "PC 7]
If :Char=-40 [Make "A :A-0.1 Make "PC 7]
If :A>5 [Make "A 5 Make "PC 1]
If :A<1 [Make "A 1 Make "PC 1]
Score :A :PC]
End

To Score :A :PC
PenUp SetPos [126 184] SetH 90 SetPC Black SetPW 10
PenDown Forward 56
PenUp SetPos [130 180] SetH 0 SetPC :PC SetPW 1
Label Sentence [a=] :A
End

To Go
New Animation Init Title Chaos
End

By varying the value of 'a', the following states can be seen.

As 'a' is decreased from 3.2 to 3.1, the plot is 'locked' either positive or negative, depending on the sign of the last plot.
Value of 'a' State
1.0 to 2.2
2.3 to 2.6
2.7
2.8 to 3.1
3.2
3.3
3.6
3.7 to 4.6
4.7to 5.0
1 state
2 state
4 state
Chaotic
Locked
6 state
4 state
Chaotic
2 state

Chaos

Procedures blue
Variables pink