Stability of fixed points of $u_{n+1}=\frac15\left(4u_n+\frac3{u_n}-2\right)$

Question

Are the fixed points of $$u_{n+1}=\frac15\left(4u_n+\frac3{u_n}-2\right)$$ where $n\in\Bbb N$, stable?

My friend and I are disagreeing but I haven't seen his work. I have obtained that the two fixed points are stable and my friend got one stable and one unstable. Who is right, if anyone?

To me, it looks like both are stable. – Ivan Neretin Oct 06 '16 at 23:00 — Ivan Neretin, Oct 06 '16 at 23:00
Thanks for your confirmation :) – Desmoz Oct 06 '16 at 23:02 — Desmoz, Oct 06 '16 at 23:02

score 2 · Accepted Answer · answered Oct 06 '16 at 23:08

2

Let $f(x) = (4x + 3/x - 2)/5$. The fixed points are the solutions of $f(x) = x$, namely $-3$ and $1$, at which $f'$ has the values $11/15$ and $1/5$ respectively. Since these have absolute value $< 1$, they are stable.

answered Oct 06 '16 at 23:08

Robert Israel

448,999

Stability of fixed points of $u_{n+1}=\frac15\left(4u_n+\frac3{u_n}-2\right)$

1 Answers1