anonymous
  • anonymous
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!
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
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?

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