Consider the series $T(n)$, defined by:
$T(0) = 0.5$
$T(n) = T(n-1) \cdot (1-(T(n-1)) \cdot k$
For values of $k$ between $1$ and $3$, as $n$ approaches infinity, $T(n)$ approaches $(1-(1/k))$. We could call this oscillation of period $1$.
For values of $k$ between $3$ and approximately $3.45$, it seems that the series settles into an oscillation of period $2$. For instance, when $k = 3.3333$, $T(n)$ ends up oscillating between about $0.83$ and $0.47$.
For values of $k$ between $3.45$ and approximately $3.56$, it seems that the series settles into an oscillation of period $4$. For instance, when $k = 3.5$, $T(n)$ oscillates between (approximately) $0.875$, $0.383$, $0.827$ and $0.501$.
For values of $k$ between $3.56$ and $3.57$, is seems to settle into an oscillation of period $8$.
Above $3.57$, the series seems quite chaotic, although it probably does eventually settle into an oscillation with a large period
Question: what are the precise values of the borders (i.e. $3$, $3.45$, $3.56$) that govern the period of the oscillation?
You have rediscovered the logistic map, which is well-known.
See https://en.wikipedia.org/wiki/Period-doubling_bifurcation