How I can plot the below map?

Question

I have tried to plot the following map:

$\qquad \begin{cases} y_{n+1}=r y_n^{x_{n+1}-x_{n+2}}\\ y_1=x_1-x_2 \quad \end{cases}$

using the code shown below. The error I have got is that

x[1] - x[2] can't be used as an iterator

However, it works if I try it analytically then how I can got its plot?

funcs = 
  RecurrenceTable[
    {y[n + 1] == r( y[n]^(x[n+1] - x[n+2])), 
     y[1] == x[1] - x[2]}, 
    y, {n, 1, 5}];
Plot[funcs /. r -> 1, {x[1] - x[2], 0, 1}, 
  Evaluated -> True,   PlotRange -> All]

Note This is just explantion about running of the below code , The below Code seems converge always to r for odd iteration and even iteration but the sequence I mean is diverge such that it take two distincts limit according the parity of iteration odd and even , As example of the above system we may take this example: $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$ here we have limit of a_n for n odd close to 0.56778606544394002098000796382530333102219963214866 and for even iteration we have 0.85885772008416606762434379473241623070938618180813 for more behavior of this sequence one can check the link of this question here in SE .Probably I'm wrong in the reformulation of the above general recursive sequence.

Answer 1

Somehow the sequence x[1], ..., x[6] must be defined. This can be done by giving each of the six values), giving a recursive formula or by defining function x over the domain [1, ..., 6]. I will use the last method to show you how a plot can be made.

x[n_] := Log2[1/(n + 1)]
Module[{r, yVals},
  r = 1;
  yVals =
    RecurrenceTable[
      {y[n + 1] == r (y[n]^(x[n + 1] - x[n + 2])), y[1] == x[1] - x[2]},
      y, {n, 1, 5}];
 ListPlot[yVals]]