Complex Dynamics of a Second Order Rational Difference Equation

TL;DR

This paper extends the analysis of a second order rational difference equation to complex parameters and initial conditions, revealing new chaotic behaviors not present in the real case.

Contribution

It investigates the dynamics of the equation with complex parameters and initial values, highlighting differences from the real case and identifying chaotic solutions.

Findings

Chaotic solutions exist for complex parameters.

Some real-line results do not hold in the complex plane.

Complex parameters lead to richer dynamical behaviors.

Abstract

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 the real parameter $\alpha$, $\beta$ and arbitrary non-negative real initial conditions is investigated a decade ago. In the present manuscript, the same has been revisited considering the parameters $\alpha$ and $\beta$ as complex numbers and the initial values as arbitrary complex numbers. It is found that some of the results which are valid in real line but does not valid in complex plane. The chaotic solutions of the difference equation with complex parameters are achieved, however there does not exists such solutions in the case of real parameters.