This paper deals with the dual incursive system of the discrete harmonic
oscillator, in the framework of discrete physics. Its basic premisses are
that nature computes incursively, and that this is a consequence of the
principle of maximum efficiency. The incursive system is based on two
parallel algorithms depending on the order in which the computations are
processed. Its incursivity, operationallity, and duality are discussed.
We study the system conceptually, analytically, numerically and
graphically. We give a number of different formulations of the equations
of motion, study the closed form solutions, shifted natural frequency of
oscillation. We find the system to be operationally efficient, orbitally
stable in phase space, and to possess constants of the motion having the
dimensions of energy.