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Let $(x_n)_{n=0}^{\infty}$ be a real sequence such that $x_0 \in (0,1)$ and for all $i \ge 1$ is $x_i = \lambda x_{i-1}(1-x_{i-1})$, where $\lambda$ is a real constant. Suppose it converges to the given limit $L$. Express $\lambda$ in terms of $L$.

I'd like to ask what's wrong with this "solution":

If $x_i \to L$, then $L=\lim_{i \to \infty} x_i = \lim_{i \to \infty}\lambda x_{i-1}(1-x_{i-1}) = \lambda L(1-L)$, solving this yields $\lambda = \frac{1}{1-L}$.

And also, how do I solve it correctly?

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  • $\begingroup$ If $\lambda =0,$ then $L=0.$ Then your formula fails. $\endgroup$
    – zhw.
    Apr 26, 2017 at 21:24

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This sequence is called Logistic map & it is an example of a discrete dynamical system with chaos. You can consider it as a dynamical system with initial condition: $$\begin{cases}x_{i+1} = f(x_i), \\ x_0 = c \in (0,1)\end{cases}$$ This system for $\lambda \in (0,4)$ has two fixed points, that is $x^{*}: f(x^{*}) = x^*$. One of these points is, obviously, 0, and the other one can be found: $$ x^* = f(x^*) = \lambda x^* (1-x^*) \Rightarrow x^* = 1 - \frac{1}{\lambda} $$ Zero is fixed point for all $\lambda$, whereas $x^* = 1 - \frac{1}{\lambda}$ is in the range (0,1) only if $\lambda \ge 1$. These fixed points for the range of $\lambda$ mentioned before are the only attractors of the dynamical system, so they are only candidates for being limits of your sequence. Now, for $x^* = 1 - \frac{1}{\lambda}$ you can express $\lambda$ in terms of $x^* \equiv L$ and it will be $\lambda = \frac{1}{1-L}$.

Deep analysis of this map and explaining what happens when $\lambda > 4$ you can find here: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536

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  • $\begingroup$ This was great, do you have a simple text on dynamical systems I could try out? I'd rather it not be applied if that's possible, otherwise I might check out the one you suggested on your post :) $\endgroup$
    – Flasgod
    Apr 26, 2017 at 22:44
  • $\begingroup$ @jorgeegroj, in fact, the book I mentioned is, probably, the simplest and the most friendly introduction to nonlinear systems, systems with chaos and other beautiful things. It has an advantage over other books covering this topic as it is written in simple language, contains detailed explanations of all statements and is fully supplied by the variety of examples and exercises. Seriously recommend it. $\endgroup$ Apr 26, 2017 at 23:39
  • $\begingroup$ okay, sounds good I'll check it out! Thanks! $\endgroup$
    – Flasgod
    Apr 27, 2017 at 16:33

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