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Consider the equaction $x=g_c(x)\equiv cx(1-x)$, with c a nonzero constant. This equation has two solutions, and we let $\alpha _c $ denote the nonzero solution. What is $\alpha _c$?For what values of c will the iteration $x_{n+1} = g_c(x_n)$ converge to $\alpha_c$ (provided that $x_0$ is chosen sufficiently close to $\alpha$)?

Can someone give me a hint how to solve this problem?

user124557
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1 Answers1

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Hint: This is the Logistic Map, which has generally a complicated behaviour. For $1<c<3$ the sequence of $x$ converge to $a_c= \frac{c-1}{c}$ which can easily be computed from $c(1-a_c)=1.\;$ For most $c > 3.56995$ there is chaotic behaviour.

gammatester
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