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I'm considering the sequence $a_{0} \in (0,1)$, $a_{n+1} = \mu a_{n}(1-a_{n})$, where $\mu \in (1,3)$ is a constant. I found that the fixed points of the function $f(x) = \mu x (1-x)$ are $0$ and $1-\frac{1}{\mu}$ and by drawing a lot of examples, I've concluded that the limit is generally $1-\frac{1}{\mu}$. However, I haven't been able to prove this.

I know that IF a limit exists, then by letting $n \to \infty$ in $a_{n+1} = \mu a_{n}(1-a_{n})$, it must be $0$ or $1-\frac{1}{\mu}$. I've tried to prove that in some cases the sequence is increasing and in others, decreasing, but the situation gets much too messy to be useful.

Is there a nice way to show that, for whichever choice of $a_{0}$, this sequence converges?

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    $\begingroup$ See en.wikipedia.org/wiki/Logistic_map#Behavior_dependent_on_r $\endgroup$
    – lhf
    May 5, 2019 at 19:32
  • $\begingroup$ $f(x)=\mu x(1-x) \Rightarrow f'(x)=\mu (1-2x)$ and $\left|f'\left(1-\frac{1}{\mu}\right)\right|=|2-\mu|<1$. This makes $x=1-\frac{1}{\mu}$ an attractive fixed point. Finding attraction interval is not so easy though. $\endgroup$
    – rtybase
    May 5, 2019 at 19:41
  • $\begingroup$ @lhf is there a proof of the claim: With r between 1 and 2, the population will quickly approach the value r − 1/r, independent of the initial population. With r between 2 and 3, the population will also eventually approach the same value r − 1/r, but first will fluctuate around that value for some time. The rate of convergence is linear, except for r = 3, when it is dramatically slow, less than linear (see Bifurcation memory). $\endgroup$ May 5, 2019 at 21:19
  • $\begingroup$ You can also try to look at the problem from the conjugate functions perspective. For example $\varphi(x)=-\mu x+\frac{\mu}{2}$ and $g(x)=x^2+\frac{\mu}{2}\left(1-\frac{\mu}{2}\right)$. It is easy to see that $f(x)=\varphi^{-1}\circ g \circ\varphi$ and $a_n=f^{\circ n}(a_0)=(\varphi^{-1}\circ g^{\circ n} \circ\varphi)(a_0)$. $\endgroup$
    – rtybase
    May 5, 2019 at 21:23
  • $\begingroup$ If I were to rephrase the question as: find the set of non-wandering points of this dynamical system, would it be easier to look at $g$ instead of $f$? Where, of course, $f(x)=\mu x (1-x)$. $\endgroup$ May 5, 2019 at 21:33

1 Answer 1

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Let $x_n = a_n -(1-\frac{1}{\mu})$. Then the recurence for $x_n$ is $$ x_{n+1} = -\mu x_n^2 + x_n(2-\mu)$$ $$\frac{|x_{n+1}|}{|x_n|} = |2 - \mu - \mu x_n| \le |2-\mu| + \mu|x_n|$$ For $\mu\in(1,3)$, $|2-\mu|<1$. So you only need to prove that at some point $|x_n| < \frac{1-|2-\mu|}{\mu}$, so that $|2-\mu| + \mu|x_n| < 1$, and from that point on $|x_n|$ will be monotonically decreasing (approximately geometrically), and will tend to $0$, that is $a_n$ will tend to $1-\frac{1}{\mu}$.

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  • $\begingroup$ $|x_{n}|<\frac{1-|2-\mu|}{\mu}$ is a pretty strong condition for $\mu$ close to $3$ or close to $1$. Do you know of any references on the logistic map for $\mu \in (1,3)$? $\endgroup$ May 5, 2019 at 22:29
  • $\begingroup$ I know no references. But it doesn't surprise me that $\mu\approx 1$ and $\mu \approx 3$ are problematic. First is $0\approx 1-\frac{1}{\mu}$ so the two stable points get close together; the other is a point of bifurcation. $\endgroup$ May 6, 2019 at 10:08

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