I have a system of 6 difference equations with three state variables. Let X_t = [x_{1t}, x_{2t}, x_{3t}, x_{4t}, x_{5t}, x_{6t}]^T. The general form of the problem is g(X_{t+1}, X_t) = 0. The log-linear form of this system around the steady state X^* is:

X_{t+1} - X^* = M \cdot (X_t - X^*)

The eigenvalues of M are -32.43, 0.56 \pm 0.94i, 0.98 \pm 0.12i and 0.94. The stability conditions of discrete time system (|\lambda|<1), which imply that there are three stable roots and the system is saddle-point stable.

If, however, I set up the system in a ‘differential form’:

X_{t+1} - X_t = (M - I) \cdot (X_t - X^*) = N \cdot (X_t - X^*)

The eigenvalues of N are -33.43, -0.44 \pm 0.94i, -0.02 \pm 0.12i and -0.06. Now, we can apply the stability conditions of differential system (Re (\lambda)<0), which implies that there are six stable roots and the system has multiple paths to the steady-state.

How do I explain the difference in the predictions of the two forms of the same model? Which is the right way to look at the problem?