When the Solutions of a System of Recurrence Relations converge I have the following system of recurrence relations:
$$a_{i+1} =  1 + z_1 a_i + z_2 b_i $$
$$b_{i+1} = 1 + z_3 a_i + z_4 b_i$$
I would like to know which conditions $z_1, z_2, z_3, z_4$ must satisfy in order for their to exist some solutions.
 A: I have never learnt the theory of generating functions, but I have studied some ODEs and the stability of dynamical systems. Intuitively, it seems as though this should work. Let me know what you think:
$$\begin{pmatrix} a_{i+1}\\\ b_{i+1} \end{pmatrix} =  \begin{pmatrix} z_1 & z_2 \\\ z_3 & z_4 \end{pmatrix}  \begin{pmatrix} a_{i}\\\  b_{i} \end{pmatrix} + \begin{pmatrix} 1\\\ 1 \end{pmatrix}$$
We can think about this in the context of systems of ODE's:
$$y' = \begin{pmatrix} z_1 & z_2 \\\ z_3 & z_4 \end{pmatrix} y + \begin{pmatrix} 1\\\ 1 \end{pmatrix}$$
Since we are only concerned with convergence, let us consider the homogeneous case:
$$x' = \begin{pmatrix} z_1 & z_2 \\\ z_3 & z_4 \end{pmatrix} x$$
Finding the eigenvalues:
$$\left( z_1 - \lambda \right) \left( z_4 - \lambda \right) - z_2 z_3 = 0$$
$$\lambda_{1,2} = \frac{z_1 + z_4}{2} \pm \frac{1}{2}\sqrt{z_1^2 + z_4^2 - 2z_1 z_4 + 4z_2 z_3}$$ 
Suppose that the contents under the square root are positive, then the system is stable if $\lambda_{1,2} < 0$ or if:
$$\frac{-z_1 - z_4}{2} >  \pm \frac{1}{2}\sqrt{z_1^2 + z_4^2 - 2z_1 z_4 + 4z_2 z_3}$$
Squaring both sides, we get:
$$z_2 z_3 < z_1 z_4 $$
Thus, in the case that $\lambda_{1,2} \in \mathbb{R}$, the system is stable if $\det \begin{pmatrix} z_1 & z_2 \\\ z_3 & z_4 \end{pmatrix} > 0$ 
This handles the case where $\lambda_{1,2}$ is real. For the imaginary case, the real part of $\lambda_{1,2}$ is simply $\frac{z_1 + z_4}{2}$. The system is stable if $z_1 + z_4 \leq 0$. 
In particular, the system is asymptotically stable if $z_1 + z_4 < 0$.
We can conclude the following: 
If $\left(z_1 - z_4 \right)^2 + 4z_2 z_3 > 0$, ($Im(\lambda) = 0$ ), then the system is stable if $z_1 z_4 - z_2 z_3 > 0$
If $\left(z_1 - z_4 \right)^2 + 4z_2 z_3 < 0$, ($Im(\lambda) \neq 0$), then the system is stable if $z_1 + z_4 < 0$
Finally, if $\left(z_1 - z_4 \right)^2 + 4z_2 z_3 = 0$, then the matrix has one eigenvalue, $\lambda = \frac{z_1 + z_4}{2}$. Clearly, the system is stable if $z_1 + z_4 < 0$
Does the stability of such a function provide insight into the convergence of these sequences ${a_i}$ and ${b_i}$?
