Abstract
This paper deals with methods to find an exact closed form solution of difference equations which are of interest in anticipatory systems. A very elementary system is the harmonic oscillator given by the differential equation

dx(t)/dt = - w² x(t)

for which several difference equation systems can be used, as for example the recursive difference equations

x(t + Dt) = x(t) + Dt v(t)
v(t + Dt) = v(t) - Dt w² x(t)

or, alternatively, the incursive difference equations

x(t + Dt) = x(t) + Dt v(t)
v(t + Dt) = v(t) - Dt w² x(t + Dt)

These latter are anticipatory because the velocity v(t + Dt) is a function of the future time step of the position x(t + Dt). When the oscillation frequency w is constant, traditional methods of solution can be used and the first system above shows an unstable solution while the second system shows a stable solution given by an orbital stability, as is the case for the solution of the differential equation. For variable frequencies, the discrete path approach has to be used in order to obtain an analytic solution. This paper will be the starting point for obtaining exact closed form solutions of more general anticipatory difference equation systems.