Review of the time-symmetric hyperincursive discrete harmonic oscillator separable into two incursive harmonic oscillators with the conservation of the constant of motion

This paper deals with a review of the properties of the hyperincursive discrete harmonic oscillator separable into two incursive discrete harmonic oscillators. We begin with a presentation step by step of the second order discrete harmonic oscillator. Then the 4 incursive discrete equations of the hyperincursive discrete harmonic oscillator are presented. The constants of motion of the two incursive discrete harmonic oscillators are analyzed. After that, we give a numerical simulation of the incursive discrete harmonic oscillator. The numerical values correspond exactly to the analytical solutions. Then we present the hyperincursive discrete harmonic oscillator. And we give also a numerical simulation of the hyperincursive discrete harmonic oscillator. The numerical values correspond also to the analytical solutions. Finally, we demonstrate that a rotation on the position and velocity variables of the incursive discrete harmonic oscillators gives rise to a pure quadratic expression of the constant of motion which is an ellipse. This result is fundamental because it gives an explanation of the effect of the discretization of the time in discrete physics. The information obtained from the incursive and hyperincursive discrete equations is richer than the information obtained by continuous physics. In conclusion, we have shown the temporal discretization of the harmonic oscillator produces a rotation similarly to the formalism of the special relativity dealing with rotations.