# Labeling Points on a Plot

I have the following discrete map: $$x_{n+1}=μ-x^4$$ for which I have used manipulate to see the way the system evolves depending on the real parameter $μ$:

Manipulate[
ListLinePlot[
NestList[μ - #^4 &, x0, 100],
PlotRange -> {0, 1},
ImageSize -> {450, 375}],
{{μ, 0.8, "parameter μ"}, 0, 4, Appearance -> "Labeled"},
{{x0, 0.2, "Initial \!$$\*SubscriptBox[\(x$$, $$0$$]\)"},
0, 1, Appearance -> "Labeled"}]

What I would like to do is to show in some way the coordinates on the plot, so that one can see the values of $x$ for which a cycle 2 is defined, a cycle 4 is defined, a cycle 8 and so on. If that is not possible, how could I for example print below the plot, the last 16 coordinates of the 100th iteration in this particular map? (so that one can see the recurrence of the 2,4,8,... points).

Thanks!

• Do you want to show all 101 coordinates on the plot, or just a select few? Would you be happy if it showed the coordinates for the points when the mouse hovers over them? Mar 10 '16 at 11:30
• I would like in particular to always show the last 16 coordinates on the plot. That would do it. Even better if I could somehow extract them and print them below the graph Mar 10 '16 at 11:32
• You are probably aware, but for $\mu$ larger than some value, the calculation aborts in an Overflow error. Mar 10 '16 at 11:37

You could add a TableForm below the plot and combine them via Column

Manipulate[Module[{list = NestList[μ - #^4 &, x0, 100]},
list2 = list;
Column[{ListLinePlot[
list,
PlotRange -> {0, 1},
ImageSize -> {450, 375}],
TableForm[Transpose@{Range[86, 101], list[[-16 ;;]]},
{{x0, 0.2, "Initial \!$$\*SubscriptBox[\(x$$, $$0$$]\)"},