Complex Dynamics of $z_{n+1}=\frac{α+ z_{n-1}}{βz_n + z_{n-1}}$

Sk Sarif Hassan

Published 2015-12-30Version 1

The dynamics of the second order rational difference equation $\displaystyle{z_{n+1}=\frac{\alpha + z_{n-1}}{\beta z_n + z_{n-1}}}$ with real parameters $\alpha$, $\beta$ and arbitrary non-negative real numbers initial conditions is investigated a decade ago. In the present manuscript, the same has been revisited considering the parameters $\alpha$ and $\beta$ are complex numbers and the initial values are arbitrary complex numbers. It is found that there are results which are valid in real set up but invalid in complex set up. There are chaotic solutions of the difference equation in the complex set up which were not in the case of real parameters.