For the Henon Map: $$ x_{n+1}=y_n+1-αx_n^2, \quad y_{n+1}=β x_n$$ I would like to discover period doubling bifurcations by varying the parameter $α$ and plotting $x_n,y_n$ versus the iterations $n$ of the map and then print below the values of the points in order to distinguish period cycles.
For one dimensional systems of the form $x_{n+1}=f(x_n)$ the period-$2$ cycles happen when the system: $$ x_1=f(x_2) \\ x_2=f(x_1) $$ has a unique solution. In other words, the trajetory "jumps" between $x_1$ and $x_2$ so while iterating the map they keep appearing (until a certain value of the parameter of the system). I have managed to do something similar for the map
$$ x_{n+1}=μ-x_n^4$$ as shown below:
Manipulate[Module[{list = NestList[μ - #^4 &, x0, 100]}, list2 = list;
Column[{ListLinePlot[list, PlotRange -> {-1, 1.5},
ImageSize -> {450, 375}],
TableForm[Transpose@{Range[86, 101], list[[-16 ;;]]},
TableHeadings -> {None, {"point", "x"}}]}]], {{μ, 0.2,
"parameter μ"}, 0, 4,
Appearance -> "Labeled"}, {{x0, 0.4,
"Initial \!\(\*SubscriptBox[\(x\), \(0\)]\)"}, 0, 1,
Appearance -> "Labeled"}]
and one can see the period two cycle (red and green are the points that repeat themselves) for a certain value of $μ$.
For a 2D system, in our case the Henon map, period-$2$ cycle means that the system: $$ 1)x_1=y_2+1-αx_2^2,\quad y_1=β x_2 \\ 2)x_2=y_1+1-αx_1^2, \quad y_2=β x_1 $$ has a unique solution and that this solution consists of two pairs of points $(x_1,y_1)$ and $(x_2,y_2)$. In this case the trajectory jumps between these two points on the plane (while at the 1D case it would be on a line).
I am actually curious how one would do it. I mean now, it has to return a pair of points each time right?
Finally, I would like to plot the Henon map after lets say $n=100$ iterations for $β=0.3$ fixed and $α$ varying until $. How can I do that?
Thank you all in advance!
EDIT Since the Henon Map is defined as above $$ x_{n+1}=y_n+1-αx_n^2, \quad y_{n+1}=β x_n$$ all I actually need is to do the same but just for the $x_{n+1}$ of the system because (and I just realized this) $y_{n+1}$ is just $β$ times the $x_n$ part! Therefore, I just want to plot $x_{n+1}$ vs $n$ and then I can deduce the value of $y_n$ but definition of the map!
I guess that is correct, right?