Synchronous Discrete Harmonic Oscillator

Abstract

We introduce the synchronous discrete harmonic oscillator, and present an analytical, numerical and graphical study of its characteristics. The oscillator is synchronous when the time T for one revolution covering an angle of 2π in phase space, is an integral multiple N of the discrete time step Δt. It is fully synchronous when N is even. It is pseudo‐synchronous when T/Δt is rational. In the energy conserving hyperincursive representation, the phase space trajectories are perfectly stable at all time scales, and in both synchronous and pseudo‐synchronous modes they cycle through a finite number of phase space points. Consequently, both the synchronous and the pseudo‐synchronous hyperincursive modes of time‐discretization provide a physically realistic and mathematically coherent, procedure for dynamic, background independent, discretization of spacetime. The procedure is applicable to any stable periodic dynamical system, and provokes an intrinsic correlation between space and time, whereby space‐discret...