Analysis of second-order recurrences using augmented phase portraits

Abstract

The augmented phase portrait, introduced by Sabrina Streipert and Gail Wolkowicz, is used to analyze second order rational discrete maps of the form \begin{align*} x_{n+1} = \frac{\alpha + \beta x_n + \gamma x_{n-1}}{A + Bx_n + C x_{n-1}}, \text{ for } n \in \mathbb{N}_0 =\{0,1,2,\dots, \} \end{align*} with parameters $\alpha, \, \beta, \, \gamma, \, A, \, B, \, C \geq 0$, and initial conditions, $x_{0}, \, x_{-1} > 0$.

First we study the special case, \begin{align*} x_{n+1} = \frac{\alpha + \gamma x_{n-1}}{A + Bx_n}, \end{align*} with $\alpha, \, \gamma, \, B > 0$ and $A \geq 0$. Applying the change of variables, $y_n = x_{n-1}$, this equation can be rewritten as a planar system. We provide a new proof to show that oscillatory solutions have semicycles of length one, except possibly the first cycle, and that nonoscillatory solutions must converge monotonically to the equilibrium. This was originally done by Gibbons, Kulenovic, and Ladas.

We also show that when the unique positive equilibrium is a saddle point, there exist nontrivial positive solutions that increase and decrease monotonically to the equilibrium, proving Conjecture 5.4.6 from the monograph by Kulenovic and Ladas. In particular, Theorem 1.2 from this monograph defines the tangent vector to the stable manifold at the equilibrium. We show that specific regions defined by the augmented phase portrait have solutions that increase and decrease monotonically to the equilibrium along the stable manifold. While Conjecture 5.4.6 was previously proven in a paper by Hoag and a paper by Sun and Xi, our proof provides a more intuitive and elementary solution.

We then consider the case, \begin{equation*} x_{n+1} = \frac{\alpha + \beta (x_n + x_{n-1})}{A + B(x_n + x_{n-1})}, \end{equation*} with $\alpha, \beta, A, B > 0$. Again, using $y_n = x_{n-1}$, this system can be written as a planar system. Thus, applying the augmented phase plane, we prove global asymptotic stability of the positive equilibrium for some cases. In other cases, we show this using other theorems from the monograph by Kulenovic and Ladas as was previously done by Atawna, et al.