Finding the Fixed Points of an Iterative Map

Question

I need to find fixed points of iterative map x[n] == 1/2 x[n-1]^2 - Mu.
My approach:

Subscript[g, n_ ][Mu_, x_] :=  Nest[0.5 * x^2 - Mu, x, n]

fixedPoints[n_] := Solve[Subscript[g, n][Mu, x] == x, x]

Plot[
  Evaluate[{x, 
   Table[Subscript[g, 1][Mu, x], {Mu, 0.5, 4, 0.5}]}
  ], {x, 0, 0.5}, Frame -> True]

Answer 1

I'll change notation slightly (mostly so I myself can understand it). You might want something like this.

y[n_, mu_, x_] := Nest[#^2/2 - mu &, x, n]
fixedPoints[n_] := Solve[y[n, mu, x] == x, x]

The salient feature is that the "function" being nested now really is a function, in correct format.

Example:

fixedPoints[2]

Out[18]= {{x -> -1 - Sqrt[-3 + 2*mu]}, 
          {x -> -1 + Sqrt[-3 + 2*mu]}, 
          {x ->  1 - Sqrt[ 1 + 2*mu]}, 
          {x ->  1 + Sqrt[ 1 + 2*mu]}}

Daniel Lichtblau

Answer 2

First of all, there is an error in your approach. Nest takes a pure function. Also I would use exact input, i.e. 1/2 instead of 0.5 since Solve is a symbolic rather than numeric solver.

Subscript[g, n_Integer][Mu_, x_] := Nest[Function[z, 1/2 z^2 - Mu], x, n]

Then

In[17]:= fixedPoints[1]

Out[17]= {{x -> 1 - Sqrt[1 + 2 Mu]}, {x -> 1 + Sqrt[1 + 2 Mu]}}

Answer 3

A side note:

Look what happens when you start very near to a fixed point (weird :) :

f[z_, Mu_, n_] := Abs[N@Nest[1/2 #^2 - Mu &, z, n] - z]

g[mu_] := f[1 + Sqrt[1 + 2*mu] - mu 10^-8, mu, 10^4]

Plot[g[mu], {mu, 0, 3}, PlotRange -> {0, 7}]  

Edit

In fact, it seems you have an autosimilar structure there: