Question

I'm really at a dead end here guys, any help is massively appreciated....
For the iteration of Z to Z^2 what effect does the starting point have?
What are the rules for behaviours of quadratic iterations?
What causes it to be chaotic, invariant, cyclic (1,2,3,n), divergent and convergent?
Basically Z is any complex number and the iteration is to square it.
So if Z0 (initial point) is 1 (1+oi) then we would sqaure it. Then sqaure that answer. Then sqaure that answer. And etc.
Obviously in that case it is invariant but I dont know what causes it to be chaotic or cyclic..
Any advice please!

Answer 1

I've experimented with a lot of different values for the initial values and have came up with some general rules for the behaviours.
Like, If zo=z1=zn it will be an invariant point.
ie. z=z^2
Then I solved Z^2 - z = 0 and found the invariant points to be 0 and 1....
Also if the modulus of z0 is < 1 it will converge and if it is > 1 it will diverge.
At this stage I am mainly trying to work out what initial values will cause the iterations to be cyclic or chaotic...
Would anyone have any advice on how to find out?