Periodic solutions of finite difference equations
The eigenvalues of the matrix A are 1 =b hi. Hence all the iterates 2/1,2/2, ■ • • , of (2) spiral away from the origin. On the other hand, every non-trivial solution of (1) is periodic with period 2ir. The essential difficulty with the differential equation (1) is that the equilibrium solution x = x = 0 is a center. Equivalently, the periodic solutions of (I) are not orbitally asymptotically stable, and thus they may be destroyed under an arbitrary small perturbation. The situation is very different, however, when x = <fi(t) (x = (a:, , ■ • • , x„)) is an orbitally asymptotically stable (unstable) solution of the differential equation