How can I compare degree of stability (or persistency) in two dynamic systems/models? Let's say I have two dynamic VAR models for two countries. What is the best way to compare persistency or degree of stability in the two models. Use eigenvalues of the two systems?
The most straightforward way is to examine the coefficients of your VAR model, since the persistence is broadly understood in econometrics as a system dependence on its own past (see Verbeek (2008): a guide to modern econometrics).  A simple VAR model is given:
$${\displaystyle y_{1,t}=c_{1}+a_{1,1}y_{1,t-1}+a_{1,2}y_{2,t-1}+e_{1,t}\,} \\
{\displaystyle y_{2,t}=c_{2}+a_{2,1}y_{1,t-1}+a_{2,2}y_{2,t-1}+e_{2,t}.\,}$$
Now, in very similar fashion as with AR models how persistent the variable is will be given by how much it depends on its own past (lags). For example, persistence of $y_1$ can be gauged by coefficient $a_{1,1}$. The larger the coefficient the more persistent the series is.