Stability of non-homogeneous and non-autonomous first-order difference equation

Question

I am seeking to analyze the stability of steady points in a system of $n$ variables $x_1(t), ..., x_n(t)$. With discrete time $t$ the system is described by

\begin{eqnarray*} x_i(t+1) = \frac{1}{t+1} \left( \left(t+C_1\right) x_i(t) - C_2\left(\sum_{j \neq i} x_j(t) \right) \right) + \frac{C_2}{t+1} \end{eqnarray*}

for each $i\in 1..n$ and with $C_1,C_2$ both real and positive.

This describes a matrix difference equation x(t+1)=A(t)x(t)+b(t), i.e., a system which is both non-homogeneous and non-autonomous. What methods exist to study stability of steady states in such systems? I am looking for any specific ideas about how to approach this problem or suggestions for further reading about stability in such systems.

Answer 1

The stability analysis of time-varying systems is a bit problematic. Here are a few sufficient conditions. The system $x(t+1) = A(t) x(t)$ is stable if any of the below conditions is satisfied:

1. $\lVert A(t) \rVert < 1, \forall t$ for some submultiplicative matrix norm.
2. There exists a $P=P^T>0$ such that $A^T(t) P A(t) - P < 0, \forall t$.
3. $\rho(A(t)) < 1, \forall t$ and there exists a $T$ such that $T^{-1} A(t) T, \forall t$ is upper triangular.
4. $\rho(A(t)) < 1, \forall t$ and $A(t_1) A(t_2) = A(t_2) A(t_1), \forall t_1, t_2$.

where $\rho(\cdot)$ is the spectral radius. Note that these are very conservative results. There are less conservative results in the literature and it is an active research topic.