There are actually, implicitly or explicitly, two different determinacy criteria in the literature: global and local determinacy.
There is global determinacy if there is only one admissible trajectory. That criterion is easy to enunciate but sometimes difficult to prove. So many people use a “local” criterion. One says that a dynamic system displays local determinacy around an equilibrium if there is only one admissible trajectory in a neighborhood of that equilibrium. That criterion will be easier to handle because, as we shall see, it can be assessed by looking only at the eigenvalues of the dynamic system.
from Benassy Macroeconomic Theory