Quadratic Function - Iteration

Iteration

Given an, one cannot always deduce the analytic form of, which means the nth iteration of . (The superscript can be extended to negative number referring to the iteration of the inverse of if the inverse exists.) But there is one easier case, in which .

In such case, one has

,

where

and .

So by induction,

can be obtained, where can be easily computed as

.

Finally, we have

,

in the case of .

See Topological conjugacy for more detail about such relationship between f and g. And see Complex quadratic polynomial for the chaotic behavior in the general iteration.