Analyzing a trajectory

Question

I have the discrete dynamical system $f(x) = 13xe^{-x}$ and want to know its stable period 4 orbit, its fixed points, and period 2 orbit. Of course also check stability of these things as well.

I've been choosing a bunch of different starting points for the period 4 orbits and it seems that for any starting point $x_0 > 0$ then eventually the period 4 orbit stabilizes to some point. However, not all starting points result in the trajectory converging to the same point.

f[x_] := 13. x E^-x bit2 = Nest[f, 1.5, #] & /@ Prepend[1]@Range[4, 400, 4] bit6 = Nest[f, 47, #] & /@ Prepend[1]@Range[4, 400, 4]

results with

{4.35104, 0.614346, 0.600992, 0.58616, 0.570842, 0.556566, 0.544936, \
0.536853, 0.532063, 0.529574, 0.528389, 0.527851, 0.527613, 0.527508, \
0.527463, 0.527443, 0.527435, 0.527431, 0.527429, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, 0.527428, \
0.527428, 0.527428, 0.527428}

-------
    {2.36701*10^-18, 5.20033*10^-15, 1.48527*10^-10, 
 4.24206*10^-6, 0.119941, 4.39942, 4.38187, 4.35943, 4.33111, \
4.29618, 4.25472, 4.20845, 4.16142, 4.11943, 4.08758, 4.06727, \
4.05616, 4.05071, 4.0482, 4.04707, 4.04658, 4.04637, 4.04627, \
4.04623, 4.04621, 4.04621, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, \
4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, \
4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, \
4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, \
4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, \
4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, \
4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, \
4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, \
4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462, \
4.0462, 4.0462, 4.0462, 4.0462, 4.0462, 4.0462}

Those are just two examples of points I tested. So what is happening? I haven't found the Stable period-$4$ orbit because the trajectories keep converging to different points for different value of starting point. Any starting point I try results in the trajectory converging to a different point though so i'm not sure what to do.

Answer 1

I think you've found two different points on the orbit of period 4, since your Range command only generates every fourth integer. I'd try this:

f[x_] = 13 x*Exp[-x];
NestList[f, 1.0, 60][[-5 ;;]]

(* Out: {0.919916, 4.76625, 0.527428, 4.0462, 0.919916} *)

Note that I've started the iteration at the critical point 1.0, since an attractive orbit must attract a critical point. I'm using NestList, rather than Nest for a bunch of different arguments. And, I'm not skipping points, so I can see the actual orbit, that is: $$0.919916 \to 4.76625 \to 0.527428 \to 4.0462 \to 0.919916.$$ Things have settled down since the [[-5;;]] construct grabs the last five terms in the orbit. I don't think it makes sense to examine just every fourth term in your orbit for then, how can you distinguish an orbit of period 4 from a fixed point or an orbit of period 2?