Recently while studying this lecture notes to learn about numerical techniques, I performed the following experiment in Mathematica as the lecture note encourage the reader to explore the logistic map (which caused me to first learn and stumble upon chaos theory in the process, hence I am still relatively new to it)
The function I am investigating is the logistic map at the most chaotic region: $$u\rightarrow 4u(1-u)$$
The surfaces were constructed by computing some kind of "difference function", which measures how much the k+1 th iterate differs from the kth iterate:
$$\text{Height}=err(u^{(k)})=4[u^{(k+1)}(1-u^{(k+1)})-u^{(k)}(1-u^{(k)})]$$
This makes $err(u^{(k)})$ to have the property that if a fixed point is present for some given u. then it will vanish for all k at that u
$err(u^{(k)})$ is then interpolated against k (increasing in the downwards direction for each panal) to give a smooth surface. The horizontal direction is u from 0 to 1 (since the chaotic region is confined here) increasing from the left
Each slab of surfaces are 5 values of k such that the first panel from the top left is $k=\text{1 to 5}$, the panel beneath is $k=\text{5 to 10}$ and so on
Interesting features are highlighted, including:
- "Transient fixed points", where the values of those u remained unchanged for a small interval of iterations
- A region of oscillation extending beyond the interval $[-1,1]$ (Hence why it got cut off)
What aspect in Chaos Theory will describe the appearance of those "transient fixed points?"
Any other features that are worth further attention and/or well described by aspects in chaos theory?
NB Mathematica code used to generate the surfaces attached for reference, adapted from this link
f3[x_] := 4*x*(1 - x)
fun = Interpolation[ Flatten[Table[{x, k, Nest[f3, x, k + 1] - Nest[f3, x, k]}, {x, 0, 1, .1}, {k, 1000, 1005, 1}], 1]];
Plot3D[fun[x, k], {x, 0, 1}, {k, 1000, 1005}, MeshFunctions -> {#2 &}, Mesh -> 10, PerformanceGoal -> "Quality", MeshStyle -> {{Black, Thin}}]