Identifying this chaotic (?) recurrence relation

Solution 1:

As far as I know, there's no hard-and-fast definition of a "true chaotic map". But there are certainly some that everybody agrees are chaotic - for example, the logistic map, with its famous bifurcation diagram.

For your map, I went ahead and generated a bifurcation diagram for the $x$-coordinates - specifically, in this graph, $k$ is along the horizontal axis, ranging from $0$ to $3$. For each value of $k$, I skipped the first $10000$ iterations, then plotted the next $10000$ $x_n$ along the vertical axis.

Those solid-black patches are chaos - even after $10000$ iterations, the $x$-coordinates were roughly evenly distributed across the interval $[0,1]$. But see those patches in the middle, where there are only a few points? Those are patches of stability, just like in the logistic map. That sort of behavior I'd call definitely chaotic.

One interesting difference from the logistic map is that it looks like it degenerates into chaos very quickly, with none or virtually none of the period-doubling characteristic of the logistic map. Or possibly the period-doubling is happening, but too fast to be seen on this scale.