Differential Equation and Stability

Question

I have an equation:

$V_{t+1}=V_t+r(S(V_t))$. r is a constant

when$(r=?)$ is $V$ asymptotically stable and when otherwise?

What I tried is, finding equilibrium points, I got: $S(V_t)=0 $ and $r=0$.

$|f'(Eq.)|$=$ $1+0$ which is not so helpful.

Any comments?

Answer 1

You're doing forward euler for the differential equation $\dot{y}(t) = S(y(t))$. So I think that maybe you're asking about A-stability and not about asymptotic stability.

To get a bound on $r$, you need to choose r such that when $S'(y(t)) < 0$, you have $|1 + r\cdot S'(y(t))| < 1$.