Dynamical system Please help!
How to show that $ \lim _{n→∞} \frac{x_{(n+1)}}{x_n} =\frac{1+\sqrt 5}{2}$ for a dynamical system 
$$x_{(n+1)}=x_n + y_n\\
y_{(n+1)}=x_n$$
Thank you!
 A: From 
$\left(\begin{array}{r} x_n \\ y_n \end{array}\right) = \left(\begin{array}{rr} 1 ~~~1 \\ 1 ~~~0 \end{array}\right)^n\left(\begin{array}{r} x_0 \\ y_0 \end{array}\right)$
you get 
$\left(\begin{array}{r} x_n \\ y_n \end{array}\right) = \left(\begin{array}{rr} F_{n+1} ~~~F_n \\ F_n ~~~F_{n-1} \end{array}\right)\left(\begin{array}{r} x_0 \\ y_0 \end{array}\right)$ 
and the rest should be clear by using the explicit formula for the Fibonacci numbers, 
e.g. have a look at the section Matrix form.
A: We should firstly shift the second equation by $1$:
 $$y_{n+1}=x_n  \rightarrow y_n = x_{n-1}$$
Let us now substitute this into the first equation:
$$x_{n+1}=x_n + y_n = x_n + x_{n-1}$$ 
Let us again shift the equation by $1$ to get:
$$x_{n}= x_{n-1} + x_{n-2}$$ 
This the recursive formula for the Fibonacci sequence where:
$$ x_n = \frac{\phi^n-(-\phi)^{-n}}{\sqrt{5}} $$
The limit now becomes:
$$ \lim _{n\to\infty} \frac{x_{n+1}}{x_n} = \lim _{n\to\infty}\frac{\frac{\phi^{n+1}-(-\phi)^{-n-1}}{\sqrt{5}}}{\frac{\phi^n-(-\phi)^{-n}}{\sqrt{5}}} = \phi = \frac{1+\sqrt 5}{2} $$
A: Let $ \lim _{n\to\infty}  x(n+1)/x(n) = a$
$\lim_{n\to\infty}  \   { x(n+1)/x(n)= \lim_{n\to\infty}  \ x(n)/x(n-1) = a} $
Given $y(n)=x(n-1)$
$x(n+1)=x(n)+x(n-1)$  (Eq. 1)
$x(n+1)/x(n)=1+x(n-1)/x(n)$
$a=1+1/a$
$a^2-a-1=0$  (Eq. 2)
$a=\frac{1+\sqrt 5}{2}$ is a solution to (Eq. 2)
