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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?

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What is $S$, or what do you know about it? –  Robert Israel Dec 2 '12 at 20:11
    
S is any differentiable function. That's all I am given. But this seems similar to eular's method for solving differential equations. Do i need this info to solve it? –  user45099 Dec 2 '12 at 20:18
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1 Answer

up vote 0 down vote accepted

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 $|1 + r\cdot S'(y(t))| < 1$.

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