Start with this Nrich activity:
Use your calculator to plug in some numbers. Start with a value for x, say x=0.2, apply your equation and get y=0.64.
Now try again, this time using this new value for x: plug in x=0.64 and get y=0.9216.
Go again, with the new value x=0.9216 and get y=0.28901376.
Carry on repeating this process (called iterating) for as long as takes your fancy, and you’ll get a sequence of numbers:
0.2, 0.64, 0.9216, 0.28901376, ________________, ________________, ________________
What do you think would have happened if you had started this process with a value of 0.2001 for x rather than 0.2? Those two numbers are very close, almost the same, so you would think they’d produce a fairly similar sequence of numbers…
____,____,_________,_____________, ________________, ________________, ________________
Try this on a spreadsheet and graph your two sequences (against sequence number). What do you notice?
Then watch https://www.youtube.com/watch?v=6z4qRhpBIyA Then try and recreate it using the code below as a starting point.
Read here about the discovery of chaos theory.
A pdf of this activity is here.