0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.