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Given the difference equation and the continuously differentiable function $g$: $$x(n+1)=x(n)+h\times g(x(n))$$ Determine conditions on $h$ for which an equilibrium point is asymptotically stable, respectively unstable.

Edward Jiang
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Anton
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1 Answers1

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Hint: From the comments, I infer that you are familiar with the requirements for stability of fixed points for the difference equation $x(n+1)=f(x(n))$. In the case you are asking about, you have $$ f(x)=x+hg(x) $$ for some function $g$ and a constant $h$. So you just need to insert this function $f$ into the criteria you already know, maybe pretty up the result a little, and you're done!

Harald Hanche-Olsen
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  • can you be more clear please? – Anton Nov 29 '14 at 21:05
  • What relevant stability criteria do you know for the case $x(n+1)=f(x(n))$? Oh never mind, you have stability for $|f'(x_0)|<1$ and instability for $|f'(x_0)|>1$, where $x_0$ is the equilibrium point, right? Now insert $f'(x_0)=1+hg'(x_0)$. – Harald Hanche-Olsen Nov 29 '14 at 21:14
  • With absolute values, yes. See my previous comment, which I amended as you were writing yours. – Harald Hanche-Olsen Nov 29 '14 at 21:19
  • I know this but do I just leave g'(x0) as it is? – Anton Nov 29 '14 at 21:19
  • Sure, what else can you do? You don't know anything more about $g$, remember. Your problem seems to be that you think the question is asking more than is reasonable. – Harald Hanche-Olsen Nov 29 '14 at 21:21
  • Ok, thanks. I had done the above reasoning myself but I thought that I had to take care of g as well. Anyway thanks for the help. – Anton Nov 29 '14 at 21:28