Stability switches in linear delay difference equations

The paper discusses necessary and sufficient conditions for the asymptotic stability of the zero solution of the linear delay difference equation y(n+1)=αy(n)+βy(n-k)y(n+1)=αy(n)+βy(n-k), where α,βα,β are complex numbers and k   is a positive integer. Compared to the case when α,βα,β are real numbers, the stability behavior of this equation turns out to be much richer. In particular, if |α|+|β|>1|α|+|β|>1 then, as k monotonously increases, the equation may switch finite times from asymptotic stability to instability and vice versa. We describe an interesting structure of the set of these stability switches, their explicit values and apply the obtained results to some important delay difference equations and their systems.