Challenging to Chaos(3)

“Prediction” means here mathematicaly as follows, with an well-known example of function F(x)=x2+a.

x1 = x02 + a
x2 = x12 + a
....
xn = xn-12 + a

What will be the value of xn under a very large value of n?
The situation will be different the two parameters: the initial value, i.e. x0 and a.

The most simple prediction can be done when

F(x)=x

Then the value stays x0 forever.
Beautiful! It would sound like snobbish, if a person as me use such a word, but it is beautiful.

Anyway such x0 for the present example can be obtained as the solution of following equation.

x02 - x0 + a = 0

Features of this solution are effected by the value of a.

Graphically this x0 can be obtained as intersections of palaboric curve y = x2 + a and y = x. Dependent on a, it might be a single solution, or nothing.

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