Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 ? :(

share|improve this question
    
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
1  
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
1  
@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
1  
$x=ax(1-x)$, 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
1  
May I suggest that someone promote a comment to an answer? Brian, Ross, Andre? –  Gerry Myerson Oct 4 '11 at 1:04
show 5 more comments

1 Answer

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.