How do I control the growth rate of this dynamical system?
$$ \left\{ \begin{aligned} &a_{n+1} = a_n + ka_n (2000 - a_n)\\ &a_0 = 4 \end{aligned} \right. $$
I experimented with multiple values of $k$ and i still can't find a way to control the growth.
(For additional context, is there a way such that this converges to a finite number and without this- ever exceeding $2000$?)
$$a_{n+1} = ka_n\left(\left(2000+\frac1 k \right) -a_n \right)$$ Define $x_n =\frac{k}{2000k+1} \cdot a_n$ then $$x_{n+1} = (2000k+1)x_n(1-x_n) \tag{1}$$
$(1)$ is the logistic map and converges to a finite value if and only if $(2000k+1) \in [0,3]$, that is $$-\frac{1}{2000} \le k \le \frac{1}{1000}$$
In this case, the sequence converges to $l \in [0,1)$ such that: $$l = \cases{0 &if $k \in \left[-\frac1 {2000}, 0 \right]$\\ 1-\frac{1}{2000k+1} &if $k \in \left[ 0,\frac{1}{1000} \right]$\\} $$