I have a system of two linear difference equations in the form:
$$ x_{1,t+2}=-Ax_{1t}-Bx_{1,t+1}-Cx_{2t}-Dx_{2,t+1}+Ex_{2,t+2}+F $$ $$ x_{2,t+2}=Gx_{2t}-Hx_{2,t+1}+Ix_{1t}+Jx_{1,t+1}-Kx_{1,t+2}+L $$
where A-L are some constants. What would be a way to solve the general solution & analyze the stability of the equilibrium state?
Thank you!
EDIT: Added specific functions.
The problem can be rewritten as
$$X(t+1)=MX(t)+N$$
for some suitable vector $X(t)$ and matrices $M$ and $N$.
The fixed points $X^*$ can be found by solving $X^*=MX^*+N$.
The stability of the equilibrium points can be studied by looking at the eigenvalues of the matrix $M$ which must be in the unit disc. Some eigenvalues can be located on the unit circle under certain conditions.