Find equilibrium points of the system
$x_{n+1}=\frac{\alpha x_n}{1+\beta x_n}$ where $\alpha>1,\beta >0$.
Are they stable?
Can someone kindly say what is meant by finding equilibrium points of the system?
I don't understand the question. I got this question in a competitive exam I sat today.
If there are any equilibrium points, they are solutions of the fixed-point equation $$ x=\frac{αx}{1+βx} $$ which can be transformed into a quadratic equation and solved as such.
One can get an explicit solution of the iteration by setting $x_n=\frac{p_n}{q_n}$ and identifying the denominators with denominators etc. to get \begin{align} p_{n+1}&=αp_n&&\implies& p_n&=α^np_0\\ q_{n+1}&=q_n+βp_n&&\implies& q_n&=q_0+β\sum_{k=0}^{n-1}α^kp_0=q_0+β\frac{α^n-1}{α-1}p_0 \end{align}