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I have been given a difference equation: $$x_{t+1} = a\cdot x_t\cdot(1-x_t)$$ and I want to find out the equilibrium points of the system. Could you recommend something to read in order to solve this problem?

The only thing I would know is to set $x_{t+1}= x_t$.

How do I solve this ? :(

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You can find a lot of fascinating properties of this iteration by searching for "Feigenbaum map". It's one of the classic examples of how chaotic behavior emerges from regular behavior under continuous parameter changes. – Henning Makholm Oct 3 '11 at 22:49
If you have stability, $x_{t+1}=x_t$; use that to get your quadratic equation. – Brian M. Scott Oct 3 '11 at 23:11
@fragant1996: taking your idea of setting $x_{t+1}=x_t$ and using $x$ for the variable, you have $x=ax(1-x)$ or $ax^2+(1-a)x=0$ with solutions $x=0,\frac{a-1}{a}$ – Ross Millikan Oct 3 '11 at 23:11
$x=ax(1-x)$, or equivalently $ax^2 -(a-1)x=0$, with the solutions $x=0$ and $x=(a-1)/a$ (if $a \ne 0$). – André Nicolas Oct 3 '11 at 23:12
May I suggest that someone promote a comment to an answer? Brian, Ross, Andre? – Gerry Myerson Oct 4 '11 at 1:04
up vote 1 down vote accepted

$x=ax(1−x)$, or equivalently $ax^2−(a−1)x=0$, with the solutions $x=0$ and $x=(a−1)/a\ $ (if $a≠0$)

Thanks @André Nicolas and @Ross Millikan

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